5" ~"X> -O
NflVflL RESEARCH
LOGISTICS
OIHBTfBLy
° - c—
I
i^r
JUNE 1981
VOL. 28, NO. 2
OFFICE OF NAVAL RESEARCH
NAVSO P-1278
NAVAL RESEARCH LOGISTICS QUARTERLY
EDITORIAL BOARD
Marvin Denicoff, Office of Naval Research, Chairman Ex Officio Members
Murray A. Geisler, Logistics Management Institute
W. H. Marlow, The George Washington University
Thomas C. Varley, Office of Naval Research
Program Director
Seymour M. Selig, Office of Naval Research
Managing Editor
MANAGING EDITOR
Seymour M. Selig
Office of Naval Research
A rlington. Virgin ia 22217
ASSOCIATE EDITORS
Frank M. Bass, Purdue University
Jack Borsting, Naval Postgraduate School
Leon Cooper, Southern Methodist University
Eric Denardo, Yale University
Marco Fiorello, Logistics Management Institute
Saul I. Gass, University of Maryland
Neal D. Glassman, Office of Naval Research
Paul Gray, Southern Methodist University
Carl M. Harris, Center for Management and
Policy Research
Arnoldo Hax, Massachusetts Institute of Technology
Alan J. Hoffman, IBM Corporation
Uday S. Karmarkar, University of Chicago
Paul R. Kleindorfer, University of Pennsylvania
Darwin Klingman, University of Texas, Austin
Kenneth O. Kortanek, Carnegie-Mellon University
Charles Kriebel, Carnegie-Mellon University
Jack Laderman, Bronx, New York
Gerald J. Lieberman, Stanford University
Clifford Marshall, Polytechnic Institute of New York
John A. Muckstadt, Cornell University
William P. Pierskalla, University of Pennsylvania
Thomas L. Saaty, University of Pittsburgh
Henry Solomon, The George Washington University
Wlodzimierz Szwarc, University of Wisconsin, Milwaukee
James G. Taylor, Naval Postgraduate School
Harvey M. Wagner, The University of North Carolina
John W. Wingate, Naval Surface Weapons Center, White Oak
Shelemyahu Zacks, State University of New York at
Binghamton
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P-35 (Revised 1-74).
APPLICATIONS OF RENEWAL THEORY IN
ANALYSIS OF THE FREE-REPLACEMENT WARRANTY*
Wallace R. Blischke
University of Southern California
Los Angeles, California
Ernest M. Scheuer
California State University, Northridge
Northridge, California
ABSTRACT
Under a free-replacement warranty of duration W, the customer is pro-
vided, for an initial cost of C, as many replacement items as needed to provide
service for a period W. Payments of C are not made at fixed intervals of
length W, but in random cycles of length Y = W + 7( WO, where y( W) is the
(random) remaining life-time of the item in service W time units after the be-
ginning of a cycle. The expected number of payments over the life cycle, L, of
the item is given by MY(L), the renewal function for the random variable Y.
We investigate this renewal function analytically and numerically and compare
the latter with known asymptotic results. The distribution of K, and hence the
renewal function, depends on the underlying failure distribution of the items.
Several choices for this distribution, including the exponential, uniform, gam-
ma and Weibull, are considered.
1. INTRODUCTION
Since a real or potential cost is involved, any item sold with a warranty must necessarily
be priced higher than if it were sold without a warranty. How much more the seller should
charge and how much more the buyer should be willing to pay depends upon the structure of
the warranty and the life distribution of the item. An analysis of pro rata and free-replacement
warranties from both buyer's and seller's points of view is given by Blischke and Scheuer [6]
and [7].
In this paper we shall consider only the free-replacement warranty and shall be concerned
mainly with the seller's (or supplier's, manufacturer's, and so forth) point of view. Of primary
importance from this point of view is the long-run profitability of the item.
An important consideration in analyzing long-run profits for items sold under a free-
replacement warranty is the expected income over the life cycle of the item. This, of course, is
a function of the expected number of replacement items sold over the life cycle. This expected
This research was supported by the Office of Naval Research under Contract No. N00014-75-C-0733, Task NR042-323
Code 434.
193
194 W. R. BLISCHKE AND E. M. SCHEUER
number, found from the renewal function for the associated random variable, is the subject of
this investigation.
In the analysis it is assumed that the buyer purchases an identical replacement when the
item in service at the end of the warranty period fails and that the purchase and initiation of
operation of a replacement are instantaneous. It is also assumed that replacements are
manufactured at the same cost and marketed at the same price. These are standard simplifying
assumptions. Though obviously unrealistic, in practice they do not negate the results of the
analysis because the important considerations are the cost/ price relativities.
Another simplifying assumption made in the analysis is that the life cycle of the item is a
constant. ("Life cycle" is also called "economic life" or "assumed life.") For planning purposes
and for tax purposes, this is, indeed, customarily taken to be a fixed quantity. In reality, of
course, equipment is purchased at different times and life cycles vary. Accordingly, the life
cycle of the item could quite properly be considered to be a random variable. This, however,
further complicates an already complex problem. Finally, it is not at all clear what might be
reasonably realistic distributional assumptions. (We know of no studies that would suggest a
particular distributional form.) Secondly, this would greatly complicate the renewal function.
It is suggested that in using the results of this paper, or any similar results, a parametric
study be done, allowing L, W, C, g, etc. (defined below) to vary over some appropriate sets of
values.
In the ensuing, we shall discuss in more detail the nature of the free-replacement war-
ranty and its associated costs/profits, the role of renewal theory in analyzing warranty policies,
and the specific renewal function encountered in the context just described.
The form of a renewal function depends ultimately on the underlying life distribution of
the items in question. Typically in dealing with renewal functions, closed form expressions are
available only for a few special cases, although limiting results are quite generally available. We
shall find this to be true of the "special" renewal function under consideration here as well.
Analytical results will be given for the exponential distribution and, to illustrate a point, the
uniform distribution. Some results of a numerical investigation of the special renewal function
for gamma and Weibull distributed lifetimes will also be discussed. These depend on a new
analytical result and on newly calculated tables (details below).
2. THE SPECIAL RENEWAL FUNCTION AND ITS ROLE IN THE
ANALYSIS OF WARRANTY POLICIES
The Analysis of Warranty Policies
In the analysis of warranty policies given by Blischke and Scheuer [6] and [7] the basic
considerations were the comparison of cost to the consumer, and of profit to the supplier, of
warranted versus unwarranted items. In the present paper, we shall limit attention to the point
of view of the supplier. From his point of view, the cost comparison leads to the establishment
of a differential pricing structure which will equate expected long-run profit in the two situa-
tions. Profit, of course, is a function of cost and income. In our previous work (Blischke and
Scheuer [6]) we derived the expected profit per warranty cycle. Here we are concerned with the
long-run profit over the life cycle of the item. This can be approximated for relatively long life
cycles by pursuing an analysis along the lines of our 1975 paper [6], (See especially Sections
2.1.1 and 2.2.) Our present objective is to obtain an exact expression for this quantity. A
result of this type would provide a basis for evaluation of the adequacy of the approximation.
RENEWAL THEORY IN WARRANTY ANALYSIS 1 95
The Free-Replacement Warranty
The specific warranty policy under consideration here is the free-replacement policy.
Under a warranty of this type the supplier provides replacements for failed items free of charge
until a specified period of service, W, is attained. His income during this period is the price, C,
charged for the initial item. His expected cost is the sum of the cost of supplying the initial
item and the expected cost of all replacements required to provide the total warranted service
time, W. In the sequel we shall express this expected cost, following Blischke and Scheuer [6],
as g[\ + Mx{ W)\, where g is the cost per unit, X is the random lifetime of an individual item
and Mx( W) is the associated renewal function evaluated at W. (In this expression the quantity
1 + MX{W) is the expected total number of items supplied; that is, the initial item plus the
expected number of replacements.)
The Excess Random Variable
For the long-run analysis of the free-replacement warranty policy, it is important to note
that no cost is incurred and no income obtained after W until the item in service at time W
fails. The symbol Y is used to denote the random time at which this event takes place. This
can also be expressed as Y = W + y(W), where y{W), the "excess random variable," is the
(random) residual lifetime of the item in service at time W. This random variable is key to the
analysis which follows. It is also called the "excess life" or "residual life" (Ross [13]), "remain-
ing life" (Barlow and Proschan [2]), and "forward recurrence time" or "residual life-time" (Cox
[9]), and has some unusual properties (see, for example, Feller [10]).
The Role of Renewal Functions
In the foregoing we have seen that the renewal function, Mx{-), of the basic lifetime ran-
dom variable, X, plays an important role in determining expected profit on a per-cycle basis. In
particular, expected profit per cycle is P = C — g[\ + Mx{ W)].
We turn now to the analysis of long-run expected profit. In this case we look at repeti-
tions of the warranty cycle. The first such cycle extends from 0 to Yx= W + y\(W), say; the
second from Y\ to Y2; and so forth. Schematically, we have
Expected
Cost g[\ + Mx(W)] 0 g[\ + Mx(W)] 0
Time
0 W
y]{W)
Income C C C ...
The total expected profit is thus seen to be P times the number of expected repetitions of this
process over the life cycle, L. This quantity is precisely the renewal function of the random
variable Y, evaluated at L. We call this the special renewal function and denote it MY(L). We
can give a closed-form expression for My(L) for X having the exponential distribution and for
L an integer multiple of W, Equation 18. Also, we can find explicitly the density and the
196 W. R. BUSCHKE AND E. M. SCHEUER
moments of y ( W) for X having the uniform distribution; however, the corresponding expres-
sion for MY(-) is not readily attainable, nor is a closed-form expression for MY(-), in general.
However, asymptotic expressions for My(L) are available and some calculations, summarized
in the portion of Section 4 showing results, indicate that a suitably chosen one of them can give
quite satisfactory approximations to My(L) over a range of L values.
In our previous work we approximated MY(L) by LI E( Y). Our present numerical inves-
tigations indicate that this does not always provide an adequate approximation. By using a new
renewal-theoretic result and with the aid of newly calculated tables we are able to obtain an
improved, and altogether quite satisfactory, approximation (see Section 4).
3. ANALYTICAL INVESTIGATION OF M) ( •)
General Renewal-Theoretic Results
We begin with the basic renewal process involving a single warranty cycle. X, Y, y(-), W
and L are as defined previously. Let Xx, X2, ... be the lifetimes of the individual items within
a warranty cycle. We assume that X\, X2, ... are nonnegative random variables which are
independent and identically distributed with cumulative distribution function Fx(-). We write
Sn = 1L *}(/!« 1,2, ■•■), So = 0, fjL = E(X), and o-2=varU). For any c.d.f., F(), we
define F - 1 - / •) = w-fold convolution of F() with itself, with [for F(0-) = 0]
<o)r,\ _
FmU)
1 t > 0
0K0.
In addition, we denote NU) = number of replacements required in the interval (0, t],
MU) = E(NU)), and mU) = M'U).
A well-known, general renewal-theoretic result is that
(1) P(N(t) = «)= F{n)U)- F{n+lHt).
This provides an immediate expression for M(t) in terms of the convolutions F{n)(t), namely
00
MU) = £ F(n)U). We turn next to the problem of determining FY() and F$n)(-).
n=\
Many asymptotic results regarding renewal functions are available. Of primary interest
here is the Elementary Renewal Theorem (Ross [13]), which was used in our previous work to
approximate MY(L). By this theorem, MYU)/t — \/E(Y) as t — ► °°. A further result, which
we will exploit in the sequel, is (Cox [9]),
(2) MyO) - J^yy + —^ - -
It has been known for some time (e.g., Smith [14]), that
(3) E(Y) = (i[l + Mx(W)).
Recently Coleman [8] has found an expression for the moments of y(W), from which the
moments of Yean be determined. In particular,
(4) var(K) = E(X2)[\ + MX(W)\ - /*2[1 + MX(W)]2 + 2/x[ WMX ■( W) - Jq Mx(u)du .
RENEWAL THEORY IN WARRANTY ANALYSIS 1 97
r w
Coleman's result, along with newly calculated tables of MX{W) and I Mx(u)du, permit the
implementation of Equation (2). These tables will be described in Section 4.
Distribution of Y
Distribution of the Excess Random Variable
Since Y = W + y ( W), the distribution of Y is simply a translation of the distribution of
the excess random variable. Thus, the fundamental result required is the distribution of y{W).
There are several ways of expressing this result. All, of course, relate back to the basic distri-
bution of X since we can also write y ( W) as y ( W) = SN x(w)+\ ~ W.
The survival function for y ( W) is given by Barlow and Proschan [2] as
(5) Fy{W)(t) = P{y{W) > t]= Fx(W+t)-fQ Fx(t +JV- u)mx{u)du.
An equivalent expression for the corresponding density is given by Cox [9] as
(6) /y(no(') = fxW+ t) + f mx(W- u)fx(u + t)du.
Mixture Representation
It is of interest to note that in addition to these classical representations, the distribution
of the excess random variable can also be expressed as a mixture of distributions (cf. Blischke
[4] and [5]), namely
oo
(7) Fy{w){t) = £ P{y{W) < t\Nx(W) = n)P{Nx{W) = n).
Here the distribution of N (given in Equation (D) is the mixing distribution and the condi-
tional distributions of y given N are the components of the mixture. Since the event
[NX{W) = n) is equivalent to the event [Sn < W, Sn+] ^ W), the conditional distributions
become
(8) P[y(W) ^ t\Nx{W) = n) = P{Sn+1 < W + t\Sn < W, S„+] ^ W),
which can be expressed as an integral over the appropriate region of the bivariate distribution of
One property often encountered in dealing with mixed distributions is that they may be
multimodal. This is indeed the case for the distribution of the excess random variable, a fact
that became quite apparent in some of our computer simulations. Another property of mix-
tures of the type we are dealing with here is that the moments of the mixed distribution can be
expressed as weighted averages of the moments of the components. We have not pursued this
point but it would be of interest in some applications. (For example, one might be interested
in the conditional expected residual lifetime of the item in service at the end of the warranty
period, given that it is the n\h replacement.)
An expression equivalent to Equation (7) is
(9) Fywb) = £ P{y(W) > t n NX(W) = n).
n=0
198 W. R. BLISCHKE AND E. M. SCHEUER
In view of the remark preceding Equation (8) and using the definition of y(HO, the joint
probabilities in Equation (9) can, for n > 1, be written
/My (HO > t H NX{W) = n) = P{Sn+] > t + W D Sn.\< W C\ Stt+] > W)
= p{sn+} > t + w n s„ < w)
(10) = f P{t + W - u ^ Sn < W\Xn+x = u}j\(u)du
(11) = J"+ [FJ^HVfO - FJtn) it + W - u))fx(u)'du
J» oo
u F{-n) (W)Mu)du.
{ + w
The limits of integration in Equation (11) come about as follows. In Equation (10) we require
t + W - u < H7, so u ^ t. Also if t + W - u < 0, i.e. u > t + W, then
(12) Pit + W - u ^ S„ < W\Xn+] = u) = P{0 ^ Sn < W\Xn+] = u)
= F^'HW),
since we are dealing with nonnegative random variables. Also
(13) /My (HO > t Pi A/v(H0 = 0) = P[XX > t + W n Xx > W)
= P[XX > t + W)
= FXU + HO.
Using Equations (11) and (13) in Equation (9), we obtain
(14) Fy(w)U) - FXU + HO + £ [/^"'(HO J*/° /*(«),
- J* Fifl)(r + W - u)fx(u)du
= FXU + W) + Mx(W)Fx(t) - f" MXU + W - u)fx(u)du.
Integrating by parts in Equation (14) and then making a change of variable in the resulting
integral yields
w
'o
which is Barlow and Proschan's formula cited at Equation (5) above.
(15) P[y{W) > t\ = f\(t + HO - f Fx(t + W - u)mx(u)du,
J 0
The density for y(HO is, from Equation (14),
(16) fy{W){,) = -jL P{y(W) > t)
l+W
= J\it + W) + J mxU + W - u)fx(u)du
which, by a change of variable of integration, is seen to be the same as Cox's formula cited at
Equation (6) above.
To complete the analysis one has to pursue the derivation of the renewal function for Y.
One approach is to translate the distribution of y( HO to obtain the distribution of Y, determine
the «-fold convolution of this distribution with itself, and hence, by Equation (1), the distribu-
tion of A/, and then determine M = E(N) directly. Exact analytical expressions can be found
by this approach only for a few special cases. In other cases the renewal function must be
approximated, either by computer simulation or by using asymptotic results. The latter
approach makes use of the Elementary Renewal Theorem or, better, of Equation (2).
RENEWAL THEORY IN WARRANTY ANALYSIS 199
Another approach to the determination of the renewal function of Y is via numerical
integration. In principle, knowledge of fx(-) permits calculation of Fx(), of the F#"'('), and
Mx{). Fy{W){-) can be obtained from (2) [numerical differentiation of Mx(-) to get mx{-) is
needed here] and then the result Fy(t) = Fy(W)(t — W) can be used. Then the successive
convolutions, F^"'(-), can be calculated, from which, finally, MY(-) can be achieved. We have
not attempted to implement this approach and know nothing about achievable accuracy or com-
puting time requirements.
Examples
The Exponential Distribution
For the exponential distribution,
ke'Kx x > 0
0 x < 0,
(17) fx(x) =
explicit expressions for all of the above are easily obtained. We use (6) to obtain the density of
the excess random variable. The "renewal density" is m(t) = 1/ E(X) = X. Thus,
(18) fyiw)(t) = \e-K{w+,) + jQ k2e-{u+,)du
= \e~kt, t > 0
which is, of course, a well-known result. The density of Y is simply a translated exponential.
The «-fold convolution of this is a translated gamma distribution, with c.d.f.
(19) FYn)(y) =
0 y < nW
(-0 '•
In writing the renewal function, it will be convenient to express L as an integer multiple of W,
say L = IW. We then obtain, from Equations (1) and (19),
(20) PiNyilW) - n) - e-*-U-»-i)w y ^'U-n-\)'W' _ -x(/-«)y y X'(/~ nYW
/=0 '• /=0 '■
n = 0,1 /- 1.
Finally, the special renewal function is found to be
(21) M)UW)= E[N}(IW)] = £ jiFl-'HlW) - F(yi+"(1W))
/=]
= fPuw) + FpHiw) + ... + F}'-])(IW)
- (i - DFi'Him
7=1 f=0 '•
200
W. R. BUSCHKE AND E. M. SCHEUER
The Uniform Distribution
Although the uniform distribution is admittedly of limited interest as a life distribution, it
is a convenient and nontrivial example to illustrate the mixture formulation. The density is
(22)
fx(x) =
0 < x < 9
9
0 otherwise.
It seems sensible to assume that 9 > W since otherwise replacements are required with proba-
bility one. However, our analysis could easily be extended to cover the case W > 9 with the
formulas presented below.
The c.d.f. of the sum of n independent uniform (0,0) random variables is
(23) FjT(x) =
for x < 0
for x > n9
n\9"
Vv -9)" + (")(* ~ 29 )2 - ... + (-1)A (£)(
for k = 0. 1 n - 1 and k9 ^ x ^ (k + 1)9.
Recalling that W < 9, we find directly that
(24) P{NX(W) = n) = F{n)(W) - F{"+])(W)
W"
n\9
n+\
9 -
W
n + 1
Also, from Equation (23) and the fact that Mx(x) can also be written as
Mx(x)= £ Fln)(x),
we find
(25)
j=o J ■
x- j9
9
j
exp
x - j9
9
, k9 < v < (A + \)9,
k = 0,1,2, ...
The density of y( W) can be shown to be
(26)
fy(W )(') =
9
w
e"
0 < t < 9 - W
1
9
w
i + w -el
- e 9 \
9 - W < t < 9
0
elsewhere.
RENEWAL THEORY IN WARRANTY ANALYSIS
It follows from this that the distribution of Y = W + y ( W) is
(27)
with mean
(28)
and variance
fyiy) =
1 -
0
u
y -0
e " - e "
E(Y)= -y e9
w_
(29) <r] = eH
0W
20*
W < y < 0
0 < y < 0 + W
elsewhere,
2 lit
+ 02-^-e» .
4
201
The above results can readily be used to express fy as a mixture. The mixing distribution
is simply the distribution of Nx, given in Equation (24). The components of the mixture are
conditional distributions, say fy(-\Nx( W) = «), of y{W) given NX(W) = n. These are found
to be
(30)
fyU\Nx(W) = n) =
1
0 -
W
W"- (W + t- 0)"
0Wn
w
n+\
0 < / < 0 - w
0 - W < t < 0.
n + \
In applications the conditional means of the excess random variable given A/A would also be of
interest. Here we find
(31)
E{y{W)\Nx(W) = n) = y-
w
1W
0
n + 1
n + 2
w
;
0 -
n + 1
The convolutions of J\ (•) are rather tedious and we have not pursued this to get a closed
expression for My(r). One could, of course, use the Elementary Renewal Theorem with (28),
or better, (2) with (28) and (29) to approximate M> (•). Finally, one might use an approach
based on the result (Barlow & Proschan [2])
(32)
A/?(s) =
F*Y(s)
1 - F*} (s)
in which * denotes Laplace-Stieltjes transform, inverting to obtain A/> (•).
The Gamma and Weibull Distributions
The gamma and Weibull distributions, with respective densities
202
W. R. BUSCHKE AND E. M. SCHEUER
(33)
and
(34)
fx(x) =
r(a))8e
xo-ie-x/p x ^ 0
-v < 0
fxix) =
-2-jc"-
/3Q
1 e"
<jr//3)°
x > 0
0
x < 0
are two of the more widely applied life distributions. Unfortunately, general, closed-form
expressions for the basic renewal functions, Afjt-(-), to say nothing of the special renewal func-
tions, My(-), exist for neither. There is, however, a closed-form expression for the basic
renewal function for the gamma distribution if the shape parameter, a, is integer-valued. (See,
for example, Barlow and Proschan [1].) The renewal density for the gamma distribution with
rational shape parameter can be obtained as well. (See Barlow and Proschan [2].) Series
expressions for the renewal function for the Weibull distribution have been given by Smith and
Leadbetter [15] and by Lomnicki [12]. Finally, the basic renewal function and other quantities
have been evaluated for certain gamma and Weibull distributions by Soland [16], for the
Weibull by White [17], for the lognormal, gamma, and Weibull by Huang [11] and for the
gamma, inverse Gaussian, lognormal, truncated normal, and Weibull by Baxter, Scheuer,
Blischke and McConalogue [3]. We will use various of these tabulations to aid us in approxi-
mating MyiL) in Section 4.
4. NUMERICAL INVESTIGATION
Structure of the Numerical Studies
Because of the complexity encountered in the analytical investigation of the distribution
of the excess random variable and the evaluation of the special renewal function, simulation
programs were written to provide an opportunity to investigate the properties of both of these
numerically. The basic life distributions that can be used in the simulations with these pro-
grams are the exponential, gamma, Weibull, uniform and normal. (The uniform for compari-
son with analytical results, the normal because of its apparent applicability in analyzing a set of
data used as an example by Blischke and Scheuer [6], and the other three because they are the
most important life distributions in the majority of applications.)
Here we shall concern ourselves only with the gamma and Weibull distributions. Some
preliminary results concerning the special renewal function for these will be discussed below.
The purpose of the special renewal program was to provide a means of investigating the approx-
imation to M){L)I L using the asymptotic expression (2) and Equations (3) and (4).
The specific results which will be reported are for the following parameter combinations
0
oc
Weibull
Gamma
2
3
4
5
1.12838
1.11985
1.10327
1.08912
.500
.333
.250
.200
RENEWAL THEORY IN WARRANTY ANALYSIS
203
These parameter combinations were initially chosen so that the tables of Soland [16] could be
used to provide numerical values for the approximation. (Soland's tables are arranged to
always have fi = 1.) Subsequently, the new tables of Baxter, Scheuer, Blischke and McConalo-
gue* became available and these were used in the calculations summarized in Tables 1 and 2,
below. All combinations of W = 0.5, 1.0 and 1.5 with L = 5, 10, and 15 were used. (This
gave warranty periods less than, equal to, and greater than the mean life and life cycles ranging
from 3+ to 30 times the warranty period.) In each simulation 500 repetitions of the special
renewal process were performed.
TABLE 1 - Values of MY(L)/L, l/E(Y),
and A(L) for the Gamma Distribution
Parameters
My(L)/L
A(L)
w
a
P
\/E(Y)
L: 5 10
15
5
10
15
0.5
2
1/2
.119
.707 .736
.753
.706
.743
.755
3
1/3
.836
.750 .794
.805
.757
.797
.810
4
1/4
.874
.792 .830
.840
.791
.833
.847
5
1/5
.902
.819 .858
.868
.817
.859
.873
1.0
2
1/2
.570
.491 .526
.539
.484
.527
.541
3
1/3
.601
.522 .561
.573
.511
.556
.571
4
1/4
.618
.533 .578
.588
.527
.572
.587
5
1/5
.628
.539 .587
.602
.536
.582
.598
1.5
2
1.2
.444
.350 .394
.413
.353
.399
.414
3
1/3
.462
.369 .410
.428
.368
.415
.430
4
1/4
.470
.377 .424
.440
.376
.423
.439
5
1/5
.476
.380 .428
.443
.380
.428
.444
TABLE 2 - Values of MY(L)/L, \/E(Y),
and A(L) for the Weibull Distribution
Parameters
My(L)lL
A(L)
w
a
0
\IE(Y)
L: 5 10
15
5
10
15
0.5
2
1.13
.845
.757 .799
.814
.761
.803
.817
3
1.12
.921
.832 .875
.888
.831
.876
.891
4
1.10
.959
.869 .912
.927
.874
.916
.931
5
1.09
.980
.882 .932
.947
.884
.932
.948
1.0
2
1.13
.616
.532 .574
.593
.525
.571
.586
3
1.12
.653
.568 .612
.625
.560
.606
.622
4
1.10
.666
.575 .622
.641
.572
.619
.635
5
1.09
.676
.581 .633
.646
.587
.632
.646
1.5
2
1.13
.469
.372 .421
.436
.373
.421
.437
3
1.12
.480
.388 .431
.446
.383
.432
.448
4
1.10
.481
.396 .431
.449
.383
.432
.448
5
1.09
.486
.397 .432
.453
.388
.437
.453
These tables give Mx (/). var[/V\ (/)], and J Mx (u)du for X having gamma, inverse Gaussian, lognormal, truncated
normal, and Weibull distributions; they encompass a broad range of parameter values and of values of /. We note that
Soland's 1968 tables do not explicitly give J MAu)du, but do include the variance of the associated equilibrium
renewal process, V At) =
2 f '
H J M\ (ii)dii, from which values of the integral can easily be obtained.
2Q4 W. R. BUSCHKE AND E. M. SCHEUER
Results
In each of the simulations the average number of renewals, say MY{L) was calculated
(along with certain additional relevant summary statistics). The basic results for the gamma
distribution are given in Table 1 and for the Weibull distribution in Table 2. In each case the
values tabulated are MY(L)/L. For comparison purposes, values of \/E( Y\ are included, as
var( Y) 1
well as values of the asymptotic approximation of -777777 +
E(Y)
2E2{Y)
= A(L).
In the simulations we also calculated the sample variances of the number of renewals for
the random variable Y. From these results one can estimate the standard error of MY(L)/L.
The results ranged from less than .002 to .009, with all standard errors except those for combi-
nations of the smallest values of W and L less than .005. Given that the accuracy of the com-
puter simulations themselves is adequate, one can therefore conclude that we have the second
digit determined to within one unit or so, except for a few cases.
Discussion
It is important to note that the approximation based on the Elementary Renewal Theorem
is somewhat inaccurate: 1/£(K) always overestimates MY(L)/L, with the difference, of
course, decreasing as L increases. (Thus L/E(Y) would consistently overestimate MY(L)
which would lead to an overestimate of the expected income over the lire cycle of the item.)
The asymptotic approximation A(L) gives quite good agreement with MY(L)/L. The
relative discrepancy between these two quantities occasionally runs up to 2%, but is mostly well
below 1%. Accordingly, it is apparent that LA(L) will generally provide a satisfactory approxi-
mation to MY(L) — certainly so in the absence of an exact mathematical expression for
MY(L) or tables of that quantity.
ACKNOWLEDGMENTS
We thank Professor R. M. Soland for bringing the thesis of his student, C. N. Huang, to
our attention and for providing us with a copy. We thank Dr. R. Coleman for making a copy of
his paper [8] available to us prior to its publication. We also thank the Editor and a referee for
helpful comments. The support of the Office of Naval Research is gratefully acknowledged.
REFERENCES
[1] Barlow, R.E. and F. Proschan, Mathematical Theory of Reliability (John Wiley and Sons,
Inc., New York, N.Y., 1965).
[2] Barlow, R.E. and F. Proschan, Statistical Theory of Reliability and Life Testing, Probability
Models (Holt, Rinehart and Winston, Inc., New York, NY., 1975).
[3] Baxter, L.A., E.M. Scheuer, W.R. Blischke and D.J. McConalogue, "Renewal Tables:
Tables of Functions Arising in Renewal Theory," technical report, School of Business
Administration, University of Southern California, to appear (1981).
[4] Blischke, W.R., "Mixtures of Discrete Distributions," in Classical and Contagious Discrete
Distributions, G.P. Patil, Editor (Statistical Publishing Society, Calcutta, India, 1965, dis-
tributed by Pergamon Press).
[5] Blischke, W.R., "Distributions, Statistical. IV. Mixtures of Distributions," in International
Encyclopedia of the Social Sciences, Vol. IV, D.L. Sills, Editor (Crowell Collier and Mac-
Millan, Inc., New York, NY., 1968).
RENEWAL THEORY IN WARRANTY ANALYSIS 205
[6] Blischke, W.R. and E.M. Scheuer, "Calculation of the Cost of Warranty Policies as a Func-
tion of Estimated Life Distributions," Naval Research Logistics Quarterly, 22, 4, 681-
696 (1975).
[7] Blischke, W.R. and E.M. Scheuer, "Application of Nonparametric Methods in the Statisti-
cal and Economic Analysis of Warranties," in The Theory and Applications of Reliability,
with Emphasis on Bayesian and Nonparametric Methods, Vol. II, C.P. Tsokos and I.N.
Shimi, Editors (Academic Press, Inc., New York, N.Y., 1977).
[8] Coleman, R., "The Moments of Forward Recurrence Time," submitted for publication.
(Copies of this paper may be obtained by writing Dr. R. Coleman, Math. Dept.,
Imperial College, London SW7 2BZ, England.)
[9] Cox, D.R., Renewal Theory (Methuen and Co., Ltd., London, 1962).
[10] Feller, W., An Introduction to Probability Theory and Its Applications, Vol. II (John Wiley
and Sons, Inc., New York, N.Y., 1966).
[11] Huang, C.N., "The Numerical Computation of Renewal Functions," Masters Thesis,
University of Texas at Austin (1972).
[12] Lomnicki, Z.A., "A Note on the Weibull Renewal Analysis," Biometrika, 53, 375-381
(1966).
[13] Ross, S.M., Applied Probability Models with Optimization Applications (Holden-Day, Inc., San
Francisco, CA, 1970).
[14] Smith, W.L., "Renewal Theory and Its Ramifications," Journal of the Royal Statistical
Society, 20B, 243-302 (1958).
[15] Smith, W.L. and M.R. Leadbetter, "On the Renewal Function for the Weibull Distribu-
tion," Technometrics, 5, 393-396 (1963).
[16] Soland, R.M., "Renewal Functions for Gamma and Weibull Dist.ibutions with Increasing
Hazard Rate," Technical Paper RAC-TP-329, Research Analysis Corp., McLean, VA
(1968).
[17] White, J.S., "Weibull Renewal Analysis," Proceedings of the Third Annual Aerospace Reliabil-
ity and Maintainability Conference, 639-657 (Society of Automotive Engineers, New
York, N.Y., 1964).
COMPARING ALTERNATING RENEWAL PROCESSES
Dalen T. Chiang
College of Business Administration
Cleveland State University
Cleveland, Ohio
Shun-Chen Niu
School of Management
University of Texas at Dallas
Richardson. Texas
ABSTRACT
Sufficient conditions are given for stochastic comparison of two alternating
renewal processes based on the concept of uniformization. The result is used
to compare component and system performance processes in maintained relia-
bility systems.
1. INTRODUCTION AND SUMMARY
Comparison of stochastic processes has been a rapidly growing area of research. In this
paper, we will study alternating renewal processes (ARP) X = {X(t), t ^ 0} where the state
space S = {0, 1) and the holding times of the process in state 1 and 0 are independent random
variables having distribution functions /-"and G. Throughout this paper, we assume F and G are
absolutely continuous with failure rate functions r(t) and <?(/), respectively. We shall denote
such a process by (X, r(t), q(t)). Similar notations will be used throughout.
Let X = \X(t), t 6 T) and Y = { Y(t), t € 7"} be two stochastic processes. We say X is
St
stochastically larger than K, denoted by X > Y, iff E f(X) ^ E f(Y) for all nondecreasing
functionals / for which the expectations exist. If X and Y have the same distribution, then we
St
write X = Y. In a recent paper, Sonderman [8] presented a set of sufficient conditions such
that stochastic comparison between two semi-Markov processes can be made. By specializing
his conditions to the case of alternating renewal processes, Sonderman (Theorem 5.1 of [8])
obtained the following result.
THEOREM 1 (Sonderman): Let (X', /-,(/), </,(/)), i= 1, 2, be two alternating renewal
processes. Assume that time 0 is a renewal point for both processes and
(a) XHO) ^ A"2(0),
(b) /-,(w) ^ r2(v),
207
208 D T CHIANG AND S. NIU
(c) q\(u) ^ Qiiv),
for all u, v > 0, then there exist two ARP's Xx and X2 defined on the same probability space
H such that X'= X\ i = 1,2, and A"1 < X2 everywhere in ft.
The purpose of this note is to show that conditions (b) and (c) in Theorem 1 can be
weakened to
(b') r\{u) ^ /-2(v) whenever u ^ v,
(c') <yi(v) < q2{u) whenever u < v.
The proof of this result and two immediate corollaries will be presented in Section 2. Section 3
contains some remarks on the main results.
2. PATHWISE COMPARISON OF ALTERNATING RENEWAL PROCESSES
We shall start by describing a construction due to Sonderman [8] which reproduces an
alternating renewal process (X, /■(/), q(t)) based on a Poisson process. In order to do that, the
following technical assumption on r(t) and q(t) is needed.
ASSUMPTION: The alternating renewal process (X, r(t), q(t)) is assumed to be unifor-
mizable, i.e., there exists a real number k < °° such that sup [r(t), </(/)} ^ k. k is called the
uniformization rate.
As discussed in Sonderman [8, pp. 113-115], this condition can be relaxed to the case
where failure rates are uniformly bounded over finite intervals. Let k be the uniformization
rate of X, the construction can be separated into two steps. First, a Poisson process with rate k
generates a sequence of potential transition epochs {/,, / ^ 0), where tQ = 0. Then a discrete
time stochastic process is constructed on {/,, / ^ 0}, determining whether each potential transi-
tion epoch is a genuine transition and, if so, the new state of the process. Specifically, let
{(Sn, 7„), n ^ 0) be a sequence of ordered pairs of integer- valued random variables, where S„
has the value 1 or 0 representing the state of the process immediately after /„. The variable
J„ = m(m < //) if the last genuine transition is at tm. We assume a genuine transition occurs
at / = 0, i.e., Jo = 0. The initial state S0 = X(0) could either be given or have an initial proba-
bility distribution. The transition probabilities of (CS„, J„), n ^ 0} are defined as:
(1) P(S„ = 0 ./„ = n I .S„ i = 1. y„_, = m, t„ i > 0) = r(t„ - tj/k
P(S„ = 1 J„ = n | S„_, = 0, /„_, = m, th i > 0) = q(t„ - tm)/k
P(S„ = S„-\, Jn = Jn-\ \Sn-\, J„-\, th / '^ 0) =
1 - PU„ = n | Vi. J„-h ',< i > 0) for 0 ^ m < n.
Finally, define a new process X = \X{t), t ^ 0} by
(2) X(t) = S„ if t„ < / < tn+x.
, SI
Then it follows from Theorem 2.1 of Sonderman [8] that X = X.
We will need the following lemma from Arjas and Lehtonen ([1], Lemma 3). See also
Theorem 3.1 of [8].
ALTERNATING RENEWAL PROCESSES 209
LEMMA 1: Let X = [X„, n ^ 0), Y = { Y„, n > 0), and Z = [Zn, n > 0} be three
discrete time stochastic processes. Suppose that
(a) (X0\Z„ = z„. h>0)^( K0 I Z„ = z„, // > 0)
and (b) (X, \X0= x0 XM = *,_,, Z„ = z„, n ^ 0) <
< Yj I ^0 = }'o Yj-\ = Jy-1. ^/, = z«. « > 0)
whenever .v, < y,, 0 ^ / ^ j — 1, for all y' ^ 1. Then there exist two stochastic processes
„ St
X = {X„, n > 0} and P = { Yn, n ^ 0} defined on the same probability space such that X — X,
« St St
Y = Y, and ,Y < Y everywhere, hence, X < Y.
We are now ready to state and prove the main theorem of this paper.
THEOREM 2: Let (X1, r,(t), q,(t)), i = 1, 2, be two uniformizable alternating renewal
processes. Assume that time 0 is a renewal point for both processes and
(a) A-'(O) < X2(0),
(b') i\(u) ^ r2(v) whenever u < v,
(c') q\(\) ^ ^2^') whenever u ^ v.
then there exist two new processes X and X defined on the same probability space H such
that X] = X\ X2 = X2 and i1 ^ X2 everywhere in H, hence Xx k X2.
PROOF: The proof is a modification of the one used by Sonderman [8] to prove his
Theorem 3.2. Since both processes are Poisson-uniformizable, let \ ^ 2 sup {/■](,), <72^)J-
The basic idea of the proof is to generate potential transition epochs for both processes by the
same Poisson process. Let {/„, n ^ 0} be a sequence of events generated by a Poisson process
with rate A. In view of Lemma 1, we need only to show that the two discrete time. stochastic
processes {S,j, n > 0} and [S2, n > 0} constructed according to (1) and (2) from Xx and X2,
respectively, satisfy the following stochastic order relationships:
(5/ I So1 = so1, .... 5,-1 = s/.!, t„, n > 0) <
(■S/ISq2 = 502 S,-i = s£i. tn, n > 0)
whenever s,-1 ^ s,2, 0 ^ /' < y — 1 for ally > 1, or equivalently,
(3) P(S/ = 1 | So1 = so1 5/-i = Sjli, t„, n > 0) ^
p(5.2 = ! | 52 = 52 S2_x = s2_h v n ^ o)
whenever 5,' ^ s,2, 0 < / < y — 1 for all y ^ 1.
Suppose (sq1 s/_i) ^ (sq, .... S/-i), and let 7,-L] = /c1 and JJl\ = /c2, where
0 < A:1 < y- 1 andO ^ k2 ^ j- 1.
210 D T. CHIANG AND S. N1U
CASE 1: Suppose s/-\ = 1, hence, sf-] = 1.
In this case, k] ^ k2 and tj — tk\ < /,- — tk2. Then by (1) and condition (b').
left hand side of (3) = I — r^itj — tki)/\ < 1 - r2(t, - tk2)/k = right-hand side of
(3).
CASE 2: Suppose s/.i = 0 and sji\ = 1.
l.h.s. of (3) = </,(/, - tki)/\ < 1/2 < 1 - r2Uj - tk2)/\ = r.h.s. of (3)
CASE 3: Suppose s,-Lj = sf-\ = 0.
In this case, kx ^ k1 and /, — rk[ ^ /, — f.2. Then from (1) and condition (c'), we have
l.h.s. of (3) = <?,(/, - rfc,)/X < <72((/ - tk2)/\ = r.h.s. of (3).
The conclusion of the theorem now follows from Lemma 1 since
S<j = A'1 (0) ^ A"2(0) = Si
Q.E.D.
The following corollaries are immediate.
COROLLARY 1: Conditions (a), (b'), and (c') in Theorem 2 can be replaced by
(i) *l(0) < A-2(0),
(ii) r\(t) or r2(t) is nonincreasing in /,
(iii) q\(f) or q2(t) is nonincreasing in t,
(iv) /•,(/) ^ r2(t) and q\(t) < q2(t) for all t > 0.
PROOF: Suppose u ^ v. If /,(/) is nonincreasing, then /,(//) > /|(v) ^ /-2(v). If
r2(t) is nonincreasing, then r\(u) > /^(w) ^ r2(v). Hence, in either case, condition (b') of
Theorem 2 is satisfied. Condition (c') can be checked in similar fashion. ~ c ^.
Q.E.D.
COROLLARY 2: Let (X, r(t), q(t)) be a uniformizable alternating renewal process.
Then there exist two alternating renewal processes (Y, />(/), </>(/)) and (Z, />(/), <//(/)),
where
rzU) — sup r(s), <//(/)= inf q(s),
/)(/)= inf r(s), qy(t) = sup q(s),
such that X is bounded stochastically from below by Z and from above by Y.
ALTERNATING RENEWAL PROCESSES 211
PROOF: Clearly the functions rz(t), Qz(t), />(/), and Q){t) are non-increasing in t.
Therefore, the conclusion is a direct consequence of Corollary 1 . n F n
3. COMMENTS AND ADDITIONS
(1) In Theorem 2, the assumption that time 0 is a renewal point for both processes can
be relaxed. It is sufficient to assume that at time 0, if both processes are in state 1, then X2 has
been in state 1 longer than X\ and if both processes are in state 0, then Xx has been in state 0
longer than X2.
(2) In a loose sense, the processes Z and Y in Corollary 2 may be viewed as the greatest
lower bound and least upper bound, respectively, for process X within the class of alternating
renewal processes whose holding times in both states are DFR (decreasing failure rate).
(3) An alternating renewal process may be used to model the performance of a repairable
component in a maintained reliability system (see [3] or Chapter 6 of [2]). The successive
operating (or repair) times of a repairable component are assumed to be independent and ident-
ically distributed random variables. All components operate independently of one another. Let
X(t) be the state of a component at time /, where
X(t) =
1 if the component is up at time /.
0 otherwise,
then X = {X(t), t > 0} is an alternating renewal process. Therefore, Theorem 2 may be used
to compare the performance of two maintained reliability systems consisting of n repairable
components. Specifically, let 0 be a coherent structure function (see [2]) and X{ ' = [X-/(t),
t ^ 0} be the performance process of the /th component in y'th systems, where / = 1,
2, . . . , n, j = 1, 2. Define XKt) = (X\ (/),..., XJ„(t)), j = 1, 2. By forming the product of
probability spaces for individual components, the following result follows directly from
Theorem 2.
PROPOSITION 1: Suppose that
(i) X,H0) < A-,2(0) for all /= 1 n.
(ii) All component performance processes are uniformizable and the failure rates satisfy
the conditions of Theorem 2.
Then there exist two stochastic processes <£' and </>2 defined on the same probability space H
such that <£' = {(f>W(0), t > 0), </>2 = \<t>{X2{t)), t ^ 0}, and & < 02 everywhere in H.
Hence, {</>(!'(')), / > 0} ^ \<i>(X2(t)), t > 0}.
(4) It is interesting to point out that an example of Miller [5, example (ii), p. 308] shows
that increasing the failure rate of downtime distribution of a component does not necessarily
increase (stochastically) the time to first system failure or system availability. Our result (see
Corollary 1) shows that for systems whose repairable components have DFR uptime and down-
time distributions, decreasing the failure rates of uptime distributions and increasing the failure
rates of downtime distributions do improve the system performance.
212 D. T. CHIANG AND S. NIU
(5) Theorem 2 may be used to establish bounds for performance measures of maintained
reliability systems. For example, one can bound the performance process of a repairable com-
ponent by that of a component whose uptime and downtime distributions are exponential (This
is a special case of Corollary 1 here or Theorem 5.1 of [8]). Maintained systems with exponen-
tial uptime and downtime distributions has been discussed in Brown [4], Ross [6 and 7]. How-
ever, the bounds obtained in this fashion are usually quite loose. Finally, we present the fol-
lowing example to illustrate the ideas involved:
EXAMPLE: Consider a two-component parallel system. Let F(G) be the uptime (down-
time) distribution of component 1 and A (/a) be the constant failure (repair) rate for component
2. Assume the system starts operation with both components new. Suppose we are interested
in the expected time until first system failure, E(T0). By conditioning on the state of the
second component when component I fails for the first time, it is not difficult to see that
E(T0) = Jo°° tclFU) + [jo°° PuU)dF(t)\ ■ E(min[D,U}) + (j0°° e~x> dG(y)] E(T0)\
where D{U) is a random variable having distribution G (exponential distribution with parame-
t) = — ^ 1 e-(A+M)/ After some simplification
K + /la A. + fi
f~ tdF(t) + [jo°° Pu(t)dF(t)\ \fQ°° e~Km-G(y))dy
ter A.) and P\\(t) = — ^ 1 e (K+fiU. After some simplification, we have
A. + /la A. + fi
/.<r0) =
■\f~ PnMdFit)]-^ e-»dG(y)
h(F,G;k,n).
Therefore, we ca bounds for E(T0) for a two-component parallel system whose first com-
ponent has the same performance process as above and the second component performance
process is unifqrmizable with failure rate function \(t) and repair rate function ix(t), t ^ 0.
Specifically, let X = sup U(/)}, A. = inf U(/)}, u = sup {u(/)|, and u. = inf {fx(t)\, then
h(F,G\k,ix) ^ E(T0) < h(F,G;,X,Jl).
REFERENCES
[1] Arjas, E. and T. Lehtonen, "Approximating Many Server Queues by Means of Single
Server Queues," Mathematics of Operations Research, J, 205-223 (1978).
[2] Barlow, R.E. and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability
Models (Holt, Rinehart, and Winston, New York, N.Y., 1975).
[3] Barlow, R.E. and F. Proschan, "Theory of Maintained Systems: Distribution of Time to
First System Failure," Mathematics of Operations Research, /, 32-42 (1976).
[4] Brown, M., "The First Passage Time Distributions for a Parallel Exponential System with
Repair," in R.E. Barlow, J.B. Fussell and N. Singpurwalla, Editors, Reliability and Fault
Tree Analysis (SIAM, Philadelphia, 1975).
[5] Miller, D.R., "A Continuity Theorem and Some Counterexamples for the Theory of Main-
tained Systems," Stochastic Processes and Their Applications, 5, 307-314 (1977).
[6] Ross, S.M., "On Time to First Failure in Multicomponent Exponential Reliability Sys-
tems," Journal of Stochastic Processes and Their Applications, 4, 167-173 (1976).
[7] Ross, S.M. and J. Schechtman, "On the First Time a Separately Maintained Parallel System
Has Been Down for a Fixed Time," Naval Research Logistic Quarterly, 26, 285-290
(1979).
[8] Sonderman, D., "Comparing Semi-Markov Processes," Mathematics of Operations
Research, 5, 110-119 (1980).
SHOCK MODELS WITH PHASE TYPE
SURVIVAL AND SHOCK RESISTANCE
Marcel F. Neuts*
University of Delaware
Newark, Delaware
Manish C. Bhattacharjee**
Indian Institute of Management
Calcutta, India
ABSTRACT
New closure theorems for shock models in reliability theory are presented.
If the number of shocks to failure and the times between the arrivals of shocks
have probability distributions of phase type, then so has the time to failure.
PH-distributions are highly versatile and may be used to model many qualita-
tive features of practical interest. They are also well-suited for algorithmic im-
plementation. The computational aspects of our results are discussed in some
detail.
1. INTRODUCTION
Shock models which relate the life distribution Hi) of a device, subject to failure by
shocks occurring randomly in time, have received considerable attention in recent years. If Pk
is the probability that the device survives k > 0, shocks and Nit) is the random number of
shocks in (0,/], the survival probability, Hi) = 1 - Hi), of such a device is given by
oo
(1) Hit)- EPNU) = £ Pk P{Nit) = k}.
The most general shock models are_ those_ that correspond to (1), such that {Nit): t > 0} is a
general counting process and 1 ^ P0 ^ P\ ^ P2 ^ . . . . Interest in and published results for
shock models center around proving that, subject to suitable assumptions on the point process
Nit) of shocks, various reliability characteristics of the shock resistance probabilities Pk are
inherited by the survival probability Hi-) in continuous time.
The first systematic treatment of such shock models was given by Esary, Marshall and
Proschan [5], when Nit) is a homogeneous Poisson process. A-Hameed and Proschan con-
sidered the cases when Nit) is a nonhomogeneous Poisson process [1] and a nonstationary
*This research was supported by the National Science Foundation under Grant No. ENG-7908351 and by the Air Force
Office of Scientific Research under Grant No. AFOSR-77-3236.
**This research was partially supported by research project 441/CMDS-APR-I at the Indian Institute of Management,
Calcutta.
213
214 M. F. NEUTS AND M. C. BHATTACHARJEE
pure birth process [2]. Block and Savits [4] treated the case when the interarrival time between
shocks is NBUE (NWUE) or NBU (NWU) and Thall [8] derived interesting, but comparatively
weaker, results when Nit) is a clustered Poisson process.
In this paper, we obtain preservation theorems for the shock model (1) when Pk is of
phase-type and so is the distribution of the interarrival time between shocks. N(t) is then a
phase type renewal process [7]. The relevance of phase type distributions (henceforth abbrevi-
ated as PH-distributions) to the algorithmic analysis of the time dependent behavior of stochas-
tic models has been discussed by Neuts in a series of papers starting with [6]. A comprehen-
sive treatment may be found in Chapter 2 of [8]. PH-distributions provide an alternative point
of departure in modelling real life distributions without the classic memoryless property and
with possible proper unimodality or multimodality. PH-distributions include the exponential,
Erlang and hyperexponential distributions as very special cases. In addition, they have the
desirable property of being closed under both finite convolutions and mixtures, a feature pos-
sessed by none of the well-known nonparametric classes of life distributions.
In Section 2, the basic properties of PH-distributions, needed in the sequel, are briefly
reviewed. The main theoretical results are discussed in Section 3. Algorithmic considerations
are presented in Section 4.
2. PH-DISTRIBUTIONS
A density \pk) on the nonnegative integers is of phase type if and only if there exists a
finite Markov chain with transition probability matrix P of order r + 1 of the form
P =
S S°
0 1
and initial probability vector [£,j8, + l], such that \pk) is the density of the time till absorption in
the state r + 1. The matrix / - S is nonsingular and the stochastic matrix .S + (1 - j3,+[)-1 ■
S ° • £ may be chosen to be irreducible.
The density [pk] is given by p0 = /3r+i, and pk = £ Sk~l S°, for k ^ 1. In this paper \pk)
will be the density of the number of shocks to failure in a reliability shock model. We will
assume throughout that /3,+1 = 0. We also clearly have that
oo
Pk = Z Pv = &Sk^ for k ^ 0.
v=k + \
The mean /a,' of [pk\ is given by£(/ - 5")_le.
A probability distribution F() on [0,°°) is of phase type if and only if there exists a finite
Markov process with generator Q of the form
Q =
-r y o
0 0
with initial probability vector [<*,«„,+ ,], such that Fi) is the distribution of the time till
absorption in the state m + 1. The matrix T is nonsingular and the generator
T + (1 - am+1)_1 J° ■ a may be chosen to be irreducible. The distribution F(-) is given by
(2) Fix) = 1 - a exp iTx)e, for x > 0.
SHOCK MODELS WITH PHASE TYPE SURVIVAL 2 1 5
We shall denote 1 — Fix) by F(x). The mean Xf of F (•) is given by \{ — — a T~x e. The pairs
(a,D and (§_,S) are called representations of F(-) and {pk} respectively. Renewal processes in
which the underlying distribution F() is of phase type were discussed in [7].
Many derivations related to PH-distributions involve the Kronecker product L <S> M of
two matrices L and M. This is the matrix made up of the blocks [Li}M\. Provided the matrix
products are defined, we have that
(3) (L ® M) {K ® H) = LK ® MH.
This property is repeately used in the sequel.
3. CLOSURE THEOREMS
We first consider the Esary-Marshall-Proschan (E.M.P.) shock model [3,5] in which
(AM?)} is a Poisson counting process of rate X.
THEOREM 1
If the number of shocks to failure has a discrete PH-density [pk,k > 0} with representa-
tion (§_,S), then the time to failure in the E.M.P. model has a continuous PH-distribution //(•)
with representation \Q,k(S — I)].
PROOF
Since Pk
= §_ Ske,
for k
>
0, we
obtain
H(t) =
fc=0
-\l
(\t)k
k\
HSke
This
proves the stated result.
H Ske = £ exp [X (S - I) t] e, for t ^ 0.
A number of interesting quantities may now be expressed in computationally convenient
forms. The >th noncentral moment of //(•) is given by
(4) ix'f = j\\-'§_(I - S)-'e, fory>l.
The density hit) = //'(/), is given by
(5) fc(f) = X£exp [k(S- I)t] S°, forr^O,
and the failure rate r(t) = h(t)H~l(t), equals
£ exp (X / S)S°
(6) r(t) = k§ — ^- ^r^, for/^0.
£ exp (X t S) e
Theorem 1 is a particular case of a more general result in which the arrivals of shocks
occur according to a PH-renewal process [7]. This result is proved next.
Let the interarrival time distribution F() be of phase type with irreducible representation
(a,T) of order m. When am+] = 1 - a e , is positive, a geometrically distributed number of
shocks occur simultaneously at each shock epoch. As in [7], we introduce the matrices P(k,t),
k ^ 0, t > 0, which satisfy the system of differential equations
216 MR NEUTS AND M. C. BH ATTACH ARJEE
(7) P'(0,t) = P(0,t)T,
P'ikj) = P(k,t)T + £ a ;;,"+', P(k -v,t)T°a, k > \,
for t > 0, with initial conditions P(k,0) = 8Qk I, for k > 0. The element P,j{k,t) is the condi-
tional probability that the Markov process with generator Q* = T + (1 - «„,+,) ' T°a, is in
the state j at time t and that k shocks have occurred in (0, t], given that it started in the state i
at time 0.
The Markov process Q* may be started according to any initial probability vector y_. With
y_= (1 — aOT + i)_,a, the PH-renewal process is started immediately after a renewal epoch.
With y_= — \[~] ar1, where \[= —a7~[e, is the mean time between shocks, we obtain the
stationary version of the PH-renewal process.
THEOREM 2
If the shocks occur according to a PH-renewal process with underlying representation
(a, T) and the process Q* is started according to the probability vector y_ and if the probability
density [pk] is of phase type with representation (^,5) of order /•, then the distribution Hi) is
of phase type with the representation
(8) k = 2®£.
K = T® / + T°a® (1 -am + ]S)-lS,
of order rm.
PROOF
By the law of total probability, we have
(9) Hit) = x Z P(k,t)e &Ske
oo
= <Z ®£) L P(kj) ® Sk (e ®e)
= (y_®d) Z(t) (e ® e), for / ^ 0.
The matrix Z(t) = £ P(k,t) ® Sk, satisfies
A = 0
oo OO
Z'(t) = £ P'(k.t) ® Sk = £ p(k,t) T® Sk
/c = 0 A: = 0
+ i I«;;'i p(k -v.t) j°a®sk
■
= Z(t) (T® 1) + £ P(k.t) T°a®Sk + ] U-am+lS)-1
k-Q
= Z(t) [T® I + T°a ® (/ - am+lS)-lS],
and clearly Z(0) = / ® 1.
SHOCK MODELS WITH PHASE TYPE SURVIVAL 2 1 7
This implies that Zit) = exp(A7), for / ^ 0. Upon substitution into (9), the proof is
complete.
Particular Cases
1. If the number of shocks to failure is geometrically distributed, i.e., Pk = 9k, for k ^ 0,
0 < 9 < 1, then
oo
(10) Hit) = 2 L P(k,t)9k e = 2exp{[T+ (1 - 0am+1)-' 0am+,r'0 JTa]/} f,
for r ^ 0.
2. In the maximum shock model, failure occurs if and only if a shock occurs whose magnitude
exceeds a critical randomized threshold Y with distribution G(-). If the magnitudes of succes-
sive shocks are independent with common distribution F (•), then
(11) Pk= f°° Fkix)dGix), for k > 0.
•J o
It follows from (10) that
(12) //(/) = J0°\exp {[7-+ (1 -am+1F(x))-1F(x)r°a]derfG(x),
for ? ^ 0, so that //(•) is a mixture of PH-distributions. If Gi) is a discrete distribution with
finite support, then //(•) itself is of phase type. Case 1 above corresponds to G() being
degenerate at 6 .
3. In the cumulative damage model, the damages are additive. With the same distributions F(-)
and Gi) as in the preceding model, we obtain
(13) Pk= \ F(k)(x)dG(x), for k > 0.
•'0
If the distribution Gi) is of phase type with representation (8,L) and X\, ... , Xk are i.i.d.
with common distribution F{), then
Pk = Jo°° Gix) dF{k)(x) = EG(XX + ... + **)
= £8 exp [LU, + ... + ^)]e = 8^fce,
where A = J exp (Lx) dF(x). It is readily seen that ,4 is a substochastic matrix of spectral
radius less than one. The density {pk} is therefore of phase type. If the_ shocks occur according
to a PH-renewal process, Theorem 2 may be applied to evaluate Hit). The matrix A is
obtained by numerical integration for general distributions f (•). If Fi) itself is of phase type
with representation (cr,/?), then
J.oo
exp iLx)a exp iRx)R °dx
o
Xoo
exp (Ix) ® exp iRx) dxil ®R")
= - (/ ®£) [i <g> / + / ® /?]-' a ®r°).
The eigenvalues of L and ^? all lie in the open left half-plane. The same then holds true for the
Kronecker sum L <8> / + / <8> R, so that the inverse exists.
The nonnegative rectangular matrix V = — iL ® / + / ® /?)"' if <S> R °), may easily be
computed by solving the system
(I ® / + / ® /?)K= - / <8> /? °
by block Gauss-Seidel iteration.
218 M. F. NEUTS AND M. C. BH ATTACHARJEE
4. ALGORITHMIC ASPECTS
We shall discuss the computation of the function //(f), which is given by Theorem 2. It
readily follows from (1) that the mean h{ of Hi) is given by A.///./, where \[ and fi{ are the
means of [pk] and F(-) respectively, whenever the PH-renewal process of arrivalsis started at a
renewal epoch. With general initial conditions, the mean h[ is given by \{ix[ + k\ — \{, where
Knowledge of the mean h{ of Hi) is useful in determining the interval over which we
wish to evaluate Hit). We may, e.g., wish to choose the mean as a convenient unit of time.
This is accomplished by replacing A' by h{K. A different rescaling may be chosen if the ele-
ments of h[K are very large or if a different time scale is desirable for the practical problem at
hand.
We now assume that the matrix K has been appropriately rescaled. The function //(/) is
computed by numerical integratrion of the system of linear differential equations
(15) x'it) = v(t)K, for t ^ 0,
v(0) = y_®@.-
and setting Hit) = v(r)f, for / > 0.
It is convenient to partition the vector xit) as W\it), .... vm(r)], where the vectors
V/(/) are /-vectors. We also set M = (/ - am+1S)"'S. The system (15) may then be rewritten
as
(16) Yj(')=£ lvU)Tvj+a,
I v,.(/) Tl
M,
for I ^ j < m. This system may be conveniently solved by a classical integration procedure,
such as Runge-Kutta. We see that the vector
I v,(f) T\
M does not depend on j and needs
to be evaluated only once in each computation of the right-hand sides of (16).
In many PU-distributions of practical interest, such as, e.g., finite mixtures of Erlang dis-
tributions, the order m of Tmay be large, but T, T° and a have very few nonzero entries. It is
then advantageous to write a special purpose subroutine to evaluate the right-hand side of (16).
By so exploiting the sparsity of T, T° and a, it is possible to reduce the computation time
greatly. The mean h{, or in general the scaling factor used in selecting the time unit, may also
be utilized to choose the step size /; in the numerical integration of the system (16). In similar
problems, we have usually made two runs at least, one with 1/50 of the time unit and one with
1/100 of the time unit. If the results at corresponding time points are not sufficiently close,
further runs with smaller steps are made. The computation times of such runs increase rapidly
and efficient programming is desirable. Other methods with a variable step size and error con-
trol may also be implemented. These classical topics in the numerical integration of ordinary
differential equations need not be belabored here. In all cases, the use of the particular struc-
ture of the matrix K is fully worthy of the additional programming effort.
BIBLIOGRAPHY
[1] A-Hameed, M.S. and F. Proschan, "Non-stationary Shock Models," Stochastic Processes
and Their Applications, /, 383-404 (1973).
SHOCK MODELS WITH PHASE TYPE SURVIVAL 2 1 9
[2] A-Hameed, M.S. and F. Proschan, "Shock Models with Underlying Birth Process," Journal
of Applied Probability, 72, 18-28 (1975).
[3] Barlow, R.E. and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability
Models (Holt, Rinehart and Winston, New York, N.Y. 1975).
[4] Block, H.W. and T.H. Savits, "Shock Models with NBUE Survival," Journal of Applied
Probability, 75, 621-628.
[5] Esary, J.D., A.W. Marshall, and F. Proschan, "Shock Models and Wear Processes," Annals
of Probability, 7, 627-649 (1973).
[6] Neuts, M.F. "Probability Distributions of Phase Type," in Liber Amicorum Professor Emer-
itus H. Florin, 173-206, Department of Mathematics, University of Louvain, Belgium
(1975).
[7] Neuts, M.F., "Renewal Processes of Phase Type," Naval Research Logistics Quarterly, 25,
445-454 (1978).
[8] Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models— An Algorithm Approach (The
Johns Hopkins University Press, Baltimore, MD (1981).
[9] Thall, P.F. "Cluster Shock Models," Tech. Rept. No. 47, University of Texas at Dallas,
Dallas, TX (1979).
AN EARLY-ACCEPT MODIFICATION TO THE TEST PLANS
OF MILITARY STANDARD 781C
David A. Butler*
Oregon State University
Corvallis, Oregon
Gerald J. Lieberman**
Stanford University
Stanford, California
ABSTRACT
This paper is concerned with the statistical test plans contained in Military
Standard 78 1C, "Reliability Design Qualification and Production Acceptance
Tests: Exponential Distribution" and the selection and use of these plans.
Modifications to the fixed-length test plans of MIL-STD-781C are presented
which allow early-accept decisions to be made without sacrificing statistical vali-
dity. The proposed plans differ from the probability ratio sequential tests in the
Standard in that rejection is permitted only after a fixed number of failures
have been observed.
1. INTRODUCTION AND SUMMARY
Military Standard 781C, "Reliability Design Qualification and Production Acceptance
Tests: Exponential Distribution" [2] covers the requirements for reliability qualification tests
(pre-production) and reliability acceptance tests (production) for equipment that experiences a
distribution of times-to-failure that is exponential. These requirements include: test condi-
tions, procedures, and various fixed-length and sequential test plans with respective
accept/reject criteria. This paper is concerned only with the statistical test plans and the selec-
tion and use of these plans. The Standard contains both fixed-length test plans (Plans IXC
through XVIIC and XIXC through XXIC) and probability-ratio sequential tests (Plans IC
through VIIIC and XVIIIC). Each fixed-length test plan is characterized by its discrimination
ratio (</), its total test time (T), and its maximum allowable number of failures to accept (k).
If a fixed-length test plan is selected, the total test duration is essentially set in advance. The
only way in which one of these plans can terminate early is by rejection. For example, Test
Plan XVIIC terminates with a reject decision at the third failure if this failure occurs before 4.3
units of total test time have transpired. An accept decision can only be made when 4.3 units of
total test time have accrued. Even if the second failure occurs very early, an early reject deci-
sion cannot be made; nor can an early-accept decision be made if no failures have occurred,
*This research was supported in part by Contract N00014-79-C-0751 with the Office of Naval Research.
'This research was supported in part by Contract N00014-75-C-0561 with the Office of Naval Research.
221
222 D. A. BUTLER AND G. J. LIEBERMAN
say, by time 4.0. In both of these situations, an early decision would lack statistical validity in
failing to guarantee the operating characteristic of the selected plan. Moreover, an early reject
decision by the consumer would probably violate confactual agreements with the producer.
However, an early-accept decision by the consumer would not be subject to such an objection.
Such a decision might seem very desirable to the consumer (government) if testing costs were
substantial or if schedule deadlines were near. This paper presents modifications to the fixed-
length test plans of MIL-STD-781C which allow early-accept decisions to be made without
sacrificing statistical validity. The proposed plans differ from the probability ratio sequential
tests in the Standard in that rejection is permitted only after a fixed number of failures have
been observed.
2. THE EARLY-ACCEPT CRITERION
The early-accept criterion we will consider is as follows. Consider a test plan 0>k with
discrimination ratio d, total test time Tk, maximum allowable number of failures to accept
k(k > 1), and consumer's risk ft. Consider alternative test plans 9(h 9\> ■■■ < &k-\ w'th tne
same discrimination ratio, maximum allowable number of failures to accept j(0 ^ ./' < A. ), and
total test times 7} = — ■ x?hb,2[/+2)> where xh-p.2j+2) 's the (1 - ft) percentile of a chi-squared
distribution with 2/ + 2 degrees of freedom.* The producer's risks for test plans
9,(0 ^ j < k) are in decreasing order of j, the test times are in increasing order of./', and the
consumer's risks are constant in j (each \s ft).
The early-accept criterion is as follows: accept at time 7}, if at most j failures have
occurred up to that time. The reject criterion remains as before: reject at the (/c + l)5' failure.
The early-accept modification alters the original test plan 9k by allowing early-accept decisions
to be made at k time points prior to the total test time Tk. As a result the producer's risk for
test plan 9k is altered. Also, even though each test plan 9$, 9\ 9k has consumer's risk
ft, and even though the alternative test plans 9o> 9\ 9k \ were only involved with accept
decisions, the consumer's risk of the resulting test is not maintained at/3, and indeed, may be
significantly greater than ft. It is true that if an early-accept decision is made at time 7}, then
test plan 9j, had it been selected prior to the start of testing, would have reached the same con-
clusion. But, by allowing the test results to effectively dictate which test plan is used, the pro-
bability calculations involved in determining the consumer's risk are modified by the condi-
tional probabilities which must consequently be incorporated into them. The producer's and
consumer's risks for the modified test plans are computed as follows. Let P A(\) denote the
probability of accepting when the true mean time between failures (MTBF) is 1/X.
k
PA(X) = X ^"{accept at time 7",}.
7=0
Let A (j) = TV {accept at time 7}}.
THEOREM 1: Suppose the true MTBF is l/\. Then
, . (\TjV exp(-\Tj) '_' ,. U(r-r/)]/-/exp(-\(r/-r/))
A(J) = T\ ~~hMl) (/-/)• •
PROOF: If an accept decision is made at time 7), then exactly / failures must have
occurred up to that time (since if fewer than / failures had occurred, an accept decision would
have been made earlier). Thus,
*This choice is somewhat arbitrary, but is motivated by the use of this rule to guarantee a given consumer's risk for a
fixed-length test plan.
EARLY-ACCEPT TEST PLANS 223
Pr {exactly j failures in [0,7^]} = Pr
\d) {accept at time 7} and
l 1=0
(j — I) failures in (7),^]}
where [o) represents a union of disjoint events.
(X Tj) ' exp(-X T.) M , , [X (2) - 7})FW exp(-X ( F, - 7)))
71 = |^(/) (T^TTi + A{J)- D
The consumer's risk for the early-accept test plan is P4(\) and the producer's risk is
1 - PA{\/d).
3. EARLY-ACCEPT TEST PLANS
It has been proposed that the early-accept criterion be used with the existing parameters
of the fixed-length test plans of MIL-STD-781C. The effect of incorporating the early-accept
criterion into these fixed-length test plans (without further modification) is shown in Table 1.
In all plans except Plan XXIC the consumer's risk is increased and the producer's risk is
decreased. (Test Plan XXIC is unchanged since it only accepts when there are no failures.)
The changes are substantial; often the consumer's risk is more than doubled and the producer's
risk halved. By altering the test time and the maximum number of failures to accept, it is pos-
sible to correct for the effect of the early-accept modification and closely match the operating
characteristics (at two points) of the standard fixed-length test plans. The corrections for each
of the MIL-STD-781C fixed-length test plans are given in Table 2. Accept times for these
early-accept test plans are listed in Table 3.
The corrections were computed by defining functions fa(T,k) as the producer's risk for
an early-accept test plan with parameters T and A:, and fp(T,k) as the consumer's risk. As T
increases fa increases and fp decreases, and as k increases fa decreases and fp increases.
Because of the integer restriction on A:, it is not always possible to design a test plan to achieve
specified values of a, /3 exactly. However, an algorithm which will determine an approximate
solution can be constructed. The algorithm from which Table 2 is derived first fixes k and uses
a quasi-Newton method to determine a value of T which will achieve the desired a -value. The
process is then repeated, varying k in accordance with a bisection search, to determine a A-value
for which /3 is also close to the desired level. Some additional checks to reduce the calculations
are also incorporated. It should be noted that the test plans of Table 2 are designed to have a
and /3 levels close to the nominal values of the standard test plans, not the actual values. (See
Tables II and C-l in [2]).
4. PERFORMANCE OF THE EARLY-ACCEPT TEST PLANS
Table 2 shows that the maximum test times for the early-accept test plans are substantially
increased from the standard test times. However, the expected test times for the early-accept
plans are much smaller than the maximum times, and compare quite favorably to the (fixed)
test times for the standard plans.* Graphs of expected test duration versus true MTBF for the
early-accept test plans appear in Figures 1-12. For comparison, the figures also graph the
expected test duration versus true MTBF for the standard test plans. The early-accept plans
*The expected test times for Early-Accept Plans IXC and XC exceed those for the corresponding standard plans for a
considerable range of the true MTBF. The reason for this is that these two early-accept plans have producer's and
consumer's risks substantially closer to the nominal values than do the standard plans.
224
D. A. BUTLER AND G. J. LIEBERMAN
TABLE 1 — Changes in Producer's and Consumer's Risks
Resulting from Incorporating Early -Accept
Criterion into MIL-STD-781C Test Plans
Without Early-Accept Option*
With Early-Accept Optiont
Test
Plan
Discrimination
Ratio
Producer's
Consumer's
Producer's
Consumer's
Risk (%)
Risk (%)
Risk (%)
Risk (%)
IXC
1.5
12.0
9.9
4.9
38.1
xc
1.5
10.9
21.4
3.5
58.8
XIC
1.5
17.8
22.1
6.8
56.4
XIIC
2.0
9.6
10.6
4.7
31.8
XIIIC
2.0
9.8
20.9
4.4
48.4
XIVC
2.0
19.9
21.0
11.3
42.8
xvc
3.0
9.4
9.9
5.9
23.1
XVIC
3.0
10.9
21.3
6.8
38.4
XVIIC
3.0
17.5
19.7
12.5
32.6
(High Risk
Plans)
XIXC
1.5
28.8
31.3
14.0
59.5
XXC
2.0
28.8
28.5
19.4
44.6
XXIC
3.0
30.7
33.3
30.7
33.3
"Taken from Tables II and III of MIL-STD-781C and is for the test plan without early-accept modification.
tTrue risk when the early-accept criterion is incorporated.
TABLE 2 -
Specifications of Standard and Early-Ac
ccpt Test Plans
Test
Discrimination
MIL-STD-781C Test Plans'*
Test
No. of Failures
Producer's Risk
Consumer's Risk
Test
No. of Failures
Plan
Ratio
Time*
to Reject
Time*
to Reject
for Corrected
Plan (%)t
for Corrected
Plan (%)t
IXC
1.5
45.0
^ 37
72.2
> 55
10.2
10.0
XC
1.5
29.9
Js 26
51.7
> 40
10.1
19.8
XIC
1 5
21.1
Ss 18
32.6
> 24
20.1
20.4
XIIC
2.0
18.8
> 14
26.0
S> 17
10.4
10.3
XIIIC
2.0
12.4
^ 10
19.1
^ 13
9.9
19.2
XIVC
2.0
7.8
> 6
12.6
> 8
20.0
18.3
xvc
3.0
9.3
> 6
12.8
> 7
10.0
8.4
XVIC
3.0
5.4
> 4
8.3
> 5
10.2
18.7
XVIIC
3.0
4.3
> 3
5.2
> 3
19.7
19.2
High Risk
Plans
XIXC
1.5
8.0
> 7
12.6
> 9
29.6
30.8
XXC
2.0
3.7
> 3
4.5
S* 3
29.9
29.1
XXIC
3.0
1 1
> 1
1.1
> 1
30.7
33.3
*In multiples of «,.
"From Tables II and III in MIL-STD-78IC
tCorrected for use with early-accept criterion to achieve irue producer's and consumer's risks close to nominal levels
as given in Table C-l of MIL-STD-781C.
EARLY-ACCEPT TEST PLANS
225
TABLE 3 — Accept Times of Early- Accept Test Plans
Test Plan
Accept Times*
IXC
To =4.2
r, = 6.1
T2 = 7.9
r3 = 9.4
r4= 11.0
r5 = 12.4
r6 = 13.9
T7= 15.3
r8 = 16.6
r9= I8.0
r.o= 19.3
Tu = 20.7
r12= 22.0
713=23.3
TX4 = 24.5
7^,5=25.8
7-,6=27.1
Tn= 28.3
7-18 = 29.6
r19= 30.8
7^= 32.1
T2\ = 33.3
r22 = 34.5
r23 = 35.8
r24 = 37.0
T2S = 38.2
7-26 = 39.4
T21 = 40.6
7-28=41.8
T29 = 43.0
7^30 = 44.2
r3I = 45.4
732 = 46.6
7-33 = 47.8
r34 = 49.0
r35=50.i
7-36=51.3
7-37=52.5
7-38 = 53.7
T39 = 54.8
T40 = 56.0
7^4, = 57.2
T42 = 58.3
r43 = 59.5
744 = 60.7
T4,= 61.8
T46 = 63.0
T41= 64.1
7-48 = 65.3
7^49 = 66.5
T50 = 67.6
7-51 = 68.8
T52 = 69.9
7-53=71.1
r54 = 72.2
XC
To =3.2
r, = 5.0
T2 = 6.6
T,= 8.1
T4 = 9.5
T, = 10.9
r6= 12.2
T7= 13.6
7-8= 14.9
T9= 16.1
7-10- 17-4
r„ = 18.7
Tl2= 19.9
7-13=21.2
r14 = 22.4
r15= 23.6
7-16=24.8
r,7-26.i
7-18 = 27.3
Tl9= 28.4
T20 = 29.6
T2\ = 30.8
r22 = 32.0
r23 = 33.2
r24 = 34.4
T2S = 35.6
7-26 = 36.7
T21 = 37.9
7-28=39.1
T29 = 40.2
7-30=41.4
7-31 = 42.5
r32 = 43.7
r33 = 44.8
7-34 = 46.0
7-35=47.1
7-36 = 48.3
T37 = 49.4
r38 = 50.6
T39= 51.7
XIC
r0=3.o
r, = 4.8
T2 = 6.3
7-3 = 7.8
T4 = 9.2
rs - 10.5
7-6= 11.9
r7 = 13.2
7-8= 14.4
7-9= 15.7
7*10- 17.0
r,,= 18.2
r,2- 19.5
7-,3 = 20.7
r14= 21.9
Tl5= 23.1
7-,6=24.3
r,7 = 25.5
7-,8 = 26.7
r19- 27.9
7-2o=29.1
T2l = 30.3
7-22=31.4
7-23 = 32.6
XIIC
7-o=3.7
r, = 5.6
r2 = 7.2
7-3 = 8.8
r4= 10.3
T5= 11.7
r6= 13.1
r7 = 14.4
r8= 15.8
r9= 17.1
7-10= 18.4
rn= 19.7
r12 = 21.0
r13= 22.3
r14= 23.5
r15 = 24.8
r16 = 26.0
XIIIC
7-o=2.8
Tx = 4.6
r2 = 6.1
7-3=7.5
r4 = 8.9
r5= 10.3
7-6= 11.6
T7= 12.9
7-8= 14.1
T9= 15.4
T10= 16.6
Tu= 17.9
r12=i9.i
XIVC
7-o=2.7
J1, = 4.4
r2 = 5.9
7-3 = 7.3
r4 = 8.7
T5 = 10.0
7-6= 11.3
T7= 12.6
xvc
r0=3.5
Ts= 11.4
r, = 5.4
r6 = 12.8
r2 = 7.0
7-3 = 8.6
t4 = 10.0
XVIC
7-0=2.5
7^ = 4.1
T2 = 5.6
T3 = 7.0
r4 = 8.3
XVIIC
7-q - 2.2
Tx = 3.8
r2 = 5.2
XIXC
7-o=2.1
r, = 3.7
r2 = 5.1
r3 = 6.4
T4 = 7.7
7-5 = 8.9
r6= 10.2
r7= 11.4
7-8= 12.6
xxc
7-0= 1.8
r, = 3.2
T2 = 4.5
XXIC
7-0= 1.1
* Accept at time Tt if y failures have occurred to that time.
226
D. A. BUTLER AND G. J. LIEBERMAN
STANDARD FIXED LENGTH TEST PLAN
WITHOUT EARLY REJECTION
STANDARD FIXEO-LENGTH TEST PLAN
WITH EARLV «£JECTION
EARLY-ACCEPT TEST PLAN
0.00 2.00 3.00 4.00
TRUE MTBF UN MULTIPLES OF LOWFR TEST MTBFI
TEST PLAN IXC
STANDARD FIXED LENGTH TEST PLAN
-WITHOUT EARLY REJECTION
STANDARD FIXED LENGTH TEST PLAN
WITH EARLY REJECTION
EARLY ACCEPT TEST PLAN
FlGI KI 1
0.00 1.00 2.00 3 00 4 00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XC
FlCii ki 2
2 18 00
g 16 00
STANDARD FIXED LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
A STANDARD FIXED LENGTH TEST PLAN
WITH EARLY REJECTION
^ EARLY ACCEPT TEST PLAN
'0 00 100 2 00 3 00 4 00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XIC
STANDARD FIXED LENGTH TEST PLAN
WITHOUT EARLY REJECTION
'STANDARD FIXED LENGTH TEST PLAN
WITH EARLY REJECTION
0 00 100 2 00 3 00 4 00 5 00 6 00
TRUE MTBF {IN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XIIC
FlGI KI 3
I ICil KI 4
m 12.00
2
rx 11 00
g
2 900
o
w 8.00
p 700
D
5 600
z
2 600
g
2 400
O
,_ 3.00
V)
2 20°
a ioo
STANDARD FIXED LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
STANDARD FIXEO LENGTH TEST PLAN
WITH EARLY REJECTION
EARLY ACCEPT TEST PLAN
0 00 100 2.00 3 00 4 00 5 00 6 00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XIIIC
STANDARD FIXED-LENGTH TEST PLAN
)00 / WITHOUT EARLY REJECTION
STANDARD FIXED LENGTH TEST PLAN
WITH EARLY REJECTION
0 00 100 2 00 3 00 4.00 5 00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XIVC
I U,l KI- 5
FlCil KI 6
EARLY-ACCEPT TEST PLANS
227
STANDARD FIXED LENGTH TEST PLAN
WITHOUT EARLY REJECTION
STANDARD FIXED-LENGTH TEST PLAN
WITH EARLY REJECTION
STANDARD FIXED LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
EARLY ACCEPT TEST PLAN
000 100 200 300 400 500 6 00 7 00 8 00 9 00
TRUE MTBF (IN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XVC
0.00
0.00 1.00 2 00
3 00 4 00 5 00 6 00 7 00 8 00 9.00
TRUE MTBF (IN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XVIC
Figure 7
Figure 8
440
4 20
4.00
380
360
340
3 20
3.00
280
2.60
240
220
200
1.80
1 60
1 40
1 20
1 00
80
20
000
STANDARD FIXED-LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
STANDARD FIXED-LENGTH TEST PLAN
WITH EARLY REJECTION
000 100 200 300 400 500 6 00 7 00 8 00 9 00
TRUE MTBF (IN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XVIIC
STANDARD FIXED-LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
; STANDARD FIXED-LENGTH TEST PLAN
WITH EARLY REJECTION
000
0.00 1 00 2.00 3.00 4.00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XIXC
Figure. 9
Figure 10
STANDARD FIXED-LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
STANDARD FIXED-LENGTH TEST PLAN
/ WITHOUT EARLY REJECTION
380
360
340
3.20
3.00
280
260
240
220
200
1.80
1 60
1.40
1 20
1 00
80
.20
0.00 .
0.00 1 00 2.00 3.00 4.00 5 00 6 00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XXC
0.00 100 2.00 3.00 4 00 600 6 00 7 00 8 00 9 00
TRUE MTBF UN MULTIPLES OF LOWER TEST MTBFI
TEST PLAN XXIC
•THE STANDARD FIXED LENGTH TEST PLAN WITH EARLY
REJECTION AND THE EARLY ACCEPT TEST PLAN ARE
IDENTICAL FOR THIS CASE
Figure 1 1
Figure 12
228
D. A. BUTLER AND G. J. LIEBERMAN
cannot be conveniently used if an estimate of the true MTBF is required. If a standard test
plan is used under these circumstances, the test continues even if a sufficient number of
failures to reject occur prior to the total test time. A graph of this plan (without early rejection)
also appears in the figures. It is not surprising that the early-accept test plans generally have
smaller expected test durations.
tion.
(1)
The expected test durations are computed as follows. Let r be the (random) test dura-
tK[Ti = t T ■ 2^, ' ' (accept at time 7";j "F '(reject in ( Tt_ ,. 7",))'
j=0
= 1 Tj-A(j) + l,Ei
7=0 ./ = 0
I- '/'
r ' A reject in (7, ,. 7,11
where IB denotes the indicator function of the event #, i.e., IH equals 1 if the event occurs, 0
otherwise. To compute the terms in the second summation in (1), note that at least j and at
most k failures must occur in [0,7}_j]. (If fewer than j failures occur, the test will accept by
time Tj-\\ if more than k failures occur, the test will reject by time 7}_i.) Given that r failures
occur in [0,7, ..J (/' ^ j ^ k) and given that the test does not terminate by time 7}_i, the test
will reject in ( 7 ,. 7}] if and only if k + 1 — r failures occur in (7}_i, 7}]. By the memoryless
property of the exponential distribution, the expected test time under these conditions is
/.
(7}_, + z)f(z)dz
where f(z) is a gamma density with parameters A and k + 1 — /. Thus, by a conditional
expectation argument
(2) Ex[r- /|rejec, in {Tj „ r>]|]-t 00- \->)f' </' , + z)/(z) A.
where Q(j,r) = Pi {do not accept or reject at or before time 7}, and r failures in [0,7,]),
(J < >).
THEOREM 2: For./ < r < k
(X 7})' exp(-X 7})
(?(/» =
a!
i' 4(1)
1=0
\\iTj- TI)Y-1 exp(-\(7; - T,))
(r- /)!
PROOF: For / < r < k,
U '
Prjexactly r failures in [0,7.]} = Pr\\£) (accept at time T,
I o
(necessarily with /failures)
and (/' - /) failures in (7), 7}])
+ Q(j.r).
(KTjVexpi-X/,)
-l
/•!
/=0
A(l)
[k(Tj- T^y-'expi-kiT-T,))
(/-/)!
+ 00.1-).
□
All that remains is to compute the integral in (2). This integral can be expressed in terms
of the incomplete gamma distribution and evaluated by standard computer subroutines [1].
EARLY-ACCEPT TEST PLANS 229
r«- <r_, + rt/w* - ro"- (7-., + z»^^ *
•»"rr,-i> (7)_, + xMe-'(x)k-'dx
Jo
T(A; - /• + 1)
Let Hx,p) be the incomplete gamma distribution, that is
Then,
X
T-T ,
/ /-I
(7)_,+2)/(z)<fe= 7}_i ■ Z(X[7}- 7}_il, A-/ + 1)
T(A—r+2) ^,7'rr/-i) e'V"'^1
= 7}_, • H\[Tj- Tj-il, k-r+l)
+ Azl±l r(X[Tj-Tj-i\, k-r+2).
5. CONCLUSIONS
From an operating characteristic curve point of view, it does not make much difference
whether a fixed-length test plan, a probability-ratio sequential test plan (truncated) or an early-
accept plan is chosen, provided each is designed to have the same operating characteristics.
Generally, the ordering of the expected test duration is smallest for the probability ratio
sequential test plan, followed by the early-accept plan, and largest for the fixed-length plan.
The advantage of the early-accept plan, over the probability ratio sequential test plan, is purely
psychological. The producer never has a lot rejected early, and early decisions occur only with
the desirable outcome (from the producer's point of view) of acceptance. Such an advantage
cannot be discounted.
ACKNOWLEDGMENT
The authors are indebted to Vlad Rutenberg for help in carrying out some of the calcula-
tions.
REFERENCES
[1] IMSL Library Reference Manual, Volume 2, International Mathematical and Statistical
Libraries, Inc., Houston, Texas, 7th Edition, February 1979.
[2] Military Standard 78IC, Reliability Design Qualification and Production Acceptance Tests:
Exponential Distribution, U.S. Department of Defense, AMSC Number 22333, 21
October 1977.
A TWO-STATE SYSTEM WITH PARTIAL
AVAILABILITY IN THE FAILED STATE
Laurence A. Baxter*
University of Delaware
Newark, Delaware
ABSTRACT
A generalization of the alternating renewal model of a repairable system to
permit partial availability in the failed slate is introduced. It is shown how, by
making use of an embedded alternating renewal process, we can readily derive
expressions for various measures of system availability Expressions for the
point availability of the generalized process are presented
1. INTRODUCTION
Consider a two-state system, i.e., a machine subject to stochastic failure and repair. If it is
assumed that the sequences of periods of operation and repair constitute an alternating renewal
process, a variety of expressions for predicting the availability of the system, known as availa-
bility measures, may be derived (see, for example, Baxter [2]). These formulae can readily be
evaluated by means of the cubic splining algorithm of Cleroux and McConalogue [4] (see also
McConalogue [5], [6]).
The model assumes that a breakdown will wholly incapacitate the system, but this need
not be the case, e.g., a large machine dependent on auxiliaries may be able to operate at a
reduced capacity if some of the auxiliaries fail. An example of such a machine is a coal-fired
boiler in which the fuel is supplied by a number of mills: while the failure of one or more of
the mills will reduce the effectiveness of the boiler, a total breakdown will not necessarily
occur. In this paper we present a generalization of the two-state system which permits partial
availability in the failed state. It will be shown that we can formulate this generalized model in
terms of an embedded alternating renewal process and hence make use of existing theory and
numerical techniques.
It is first necessary to introduce some notation. Let F and G denote the distribution func-
tions of the failure and repair times respectively and suppose that these have finite expectations
and variances fx{, /jl2, ctj2, and or22, respectively. Define the indicator variable of the two-state
system
1 if the system is operating at /
k 0 otherwise
'This research was performed while the author was at University College. London, England.
231
232 L. A. BAXTER
where k = 0(1) if the system enters the down (up) state at f = 0. We define the Stieltjes con-
volution of two functions, P and Q say, each with support on the nonnegative real line as
P * Qit) = f Pit - u) dQ{u)
'0
and the «-fold recursive convolution of Pit) is denoted P{n)(t). The point availability of the
two-state system is defined as Akit) = p{Ikit) =1}. It can be shown that
(1) Ai(t)= Fit) + F * Hit)
(2) A0(t) = Git) - G * Hit)
where
(3) Hit)= £ FM * G{n)it)
denotes the renewal function of the sequence of failures (repairs) embedded in the alternating
renewal process if there is a failure (repair) at / = 0 and where Pit) = 1 - Pit) for any func-
tion Pit) such that 0 ^ Pit) ^ 1 for all t (see, for example, Baxter [2]).
2. THE GENERALIZED MODEL
There are many ways of generalizing the alternating renewal model to allow for partial
availability in the failed state. We could, for example, assume n levels of partial availability and
hence generalize the two-state system to an in + l)-state semi-Markov process. This would
result in a considerably more complex model for a relatively little increase in generality.
The approach adopted here is to assume that a proportion y, (0 < y ^ 1), of break-
downs exhibit partial availability and that the level, A, is a random variable, independent of the
failure and repair times, with distribution function M. The value of A is assumed to remain
constant during any given period of repair. The distribution M is conditional on { \ > 0}
(although we could equally consider a distribution which assigns a mass of probability 1 - y to
the value 0). This model is equivalent to a three-state semi-Markov process with transition
matrix
Available
Partially available
Wholly unavailable.
We now define the multistate variable
0
y
1
- y
1
0
0
1
0
0
7(r) =
1 if the system is fully available at /
X if the system is only operating at level A (0 < X < 1 ) at /
0 if the system is wholly unavailable at /.
In particular, [Jkit), t > 0} denotes the generalized process in which there is a failure (repair)
at t - Oif k= 0(1).
A variety of types of availability measure can be defined for the process {Jit), t > 0}.
We could, for example, consider the expectation; in particular
(4) E[JiU)}" AXU) +y £(A) AXU)
(5) E[J0(t)}= A0it) +y £(A) A0it).
TWO-STATE SYSTEM WITH PARTIAL AVAILABILITY
233
Similarly, the expected proportion of time for which the system is wholly or partially available
in iO,t] is given by
(6)
(7)
= fl,(/) +yE(A) a,(f)
t Jo '
- JQ' JQiu)du\ = a0it) + y EiA) a0U)
1 C '
where ak{t) = — J Akiu)du denotes the average availability of the process {lk(t), t > 0}
([1], p. 191).
EXAMPLE
Consider the alternating Poisson process, i.e., Fit) = 1 — e "' and Git) = 1 — e M'; in
this case it is well known that
A At)
A0U)
_ + v e-(" +M>'
V + fl V + fl
(*■
V + (X V + fl
_ e-(y +M)'
Suppose that A ~ Beta (a,/3) and hence E(\) = a /(a +j8). Substituting these expres-
sions into (4) and (5) gives us the following formulae for ElJ^t)} and E[J0(t)}:
Fl l (t)) = M« + M/3 + ayv , via + (3 - ay) -(,+,,),
1 ' ,] (v+fi) ia+(3) iv+ix) ia+p)
F\ I (,)) = M« + M^ + ajv _ ti(a + P - ay) p-{v\^t
1 °U;,~ (v+n) (a+fi) (v+n)(a+p)
a
3. THE AUGMENTED PROCESS
The expectation of the multistate variable is of limited use as there is no obvious exten-
sion to, for example, interval availability, i.e., p[Ht) = 1 V / 6 [fi,^]) [2]. Further, this
measure is not very sensitive to the distribution of A; only £(A) is required and hence identi-
cal forecasts would result from two distributions with the same mean but different variances.
Observe also that, for y > 0, E[Jit)} > A it) and hence E
)du
> ait). Thus, any
positive value of A, no matter how small, increases the measure of system availability. This
could be an unrealistic assumption in practice: if A is close to 0, it may not be worthwhile
attempting to use the machine until it is fully repaired.
An alternative approach, which is more likely to be of use in practice, is to regard the sys-
tem as operating satisfactorily if A > X0 and as broken down otherwise. The system can thus
undergo an arbitrary number of changes of state without becoming unavailable provided that
each repair period exhibits partial availability at a level exceeding KQ. The alternating sequence
of periods of availability at a level no less than \0 and periods of repair in which A < \0 clearly
constitutes an embedded alternating renewal process for fixed X0- Thus, if we define
lit)
1 if Jit) > \0
0 otherwise
234 L A BAXTER
we can apply the arguments of Baxter [2] to derive expressions for probabilities of the form
p{J(t) > X0 V t € r}, where 7 is an index set comprising an arbitrary (finite) series of points
and intervals, for the process {/(/), / > 0}, which we call the augmented process. In particular,
we shall consider {lk(t), t > 0}, the augmented process in which there is a failure with A < \0
(repair) at t = 0 for k = 0(1). It is important to appreciate that the interpretation of the sub-
script k is not the same for functions defined with respect to the two-state system and those
defined with respect to the augmented process. For the former, the values 0 and 1 are used to
denote a failure and repair at / = 0 respectively, whereas for the latter, these values denote a
failure at t = 0 such that the level of partial availability during the succeeding downtime is less
than \0 and a repair at / = 0, respectively. If y = 0, the augmented process degenerates to the
two-state system and the interpretations of the two subscripts coincide.
Let E denote the duration of an "uptime" in the augmented process, i.e., the time from a
repair following a downtime with A < \0 to the beginning of the next such downtime, and sup-
pose that this has distribution function <£>. It is easily seen that
(8) 4>(/)= (1 -a) £ a" F{n + U * Gin) (t)
where a = yp{\ > \Q} = y M(\0) and where
G,0,(/) =
1 if t > 0
0 if t < 0.
We can readily derive expressions for the mean and variance of E by means of conditional
expectation:
(9) *<B)-"1+fl^
1 -
a
(10) var(E)= — ^— (<r ,2 + o-22) + ^-rrWi +/z2)2+o-,2.
I - a (1 - nt)z
Observe that if a = 1, both mean and variance are infinite. This is to be expected as in this
case the system is always available. Similarly, if a = 0, the augmented process reduces to the
alternating renewal model and £(E) = m h var (E) = <r j2.
EXAMPLE
Consider the alternating Poisson process. The Laplace-Stieltjes transforms of Fand G are
given by f*(s) = v/{s + u) and g*(s) = (jl/(s + fi) respectively, and hence the Laplace-
Stieltjes transform of 4> is
v (1 —a) (s + /i.)
4>*(s) =
(s + v) (s + fj.) — au/x
= i/(l -a)
M
(s + A) (s + B) (s + A) (s + B)
where A,B = y [-(*> + /x) ± >/{(«/ +/i)2- 4vfi (1 - a)}].
Thus, on inversion, we see that the density of E is
<M'>= Vi}~aj lAe-Al- Be-B'-v.(e-A'-e-Bf)].
A — B
TWO-STATE SYSTEM WITH PARTIAL AVAILABILITY 235
Observe that <f)(t) is a special case of the density of the first passage time to absorption in the
Chiang-Hsu alternating renewal process with an absorbing state [3].
4. POINT AVAILABILITY
The point availability A it) = p{lit) = 1} of the augmented process is the probability that
the system is available at / or that it is under repair and that the level of partial availability
exceeds A0- The following expressions for Axit) and A0it) are obtained by substituting
, . (i -a)r(s)
* \-af*(s)g*(s)
and g*(s) into the formulae for Afis) and Aftis), performing some rearrangement and invert-
ing:
(11) Ax(t) = AXU) +aAl(t)
(12) A0U)= (1 -«) ^o(') +« GU).
As would be expected, Akit) = Akit) if a = 0 (k = 0, 1) as in this case {lkit), t > 0} reduces
to [lkit), t > 0). Similarly, if a = 1 the system cannot fail and hence Ax(t) = 1 and
A0(t)= Git).
Expression (11) clearly corresponds to (4) whereas expression (12) does not correspond
so obviously to (5); an interpretation of this result is, however, more evident if we make use of
(2) to rewrite (12) as
(13) A0U)= A0(t) +aG * Hit).
We now see that we are increasing A0it), the point availability of the two-state system, by the
probability that the system fails at u < t and that the succeeding repair time, which exhibits
partial availability at a level exceeding \0, is greater than / — w, for each u € i0,t].
EXAMPLE
On substituting the formulae for the point availabilities of the alternating Poisson process
into (11) and (12) we obtain the following expressions for the point availabilities of the
corresponding augmented process:
V + /JL V + /JL
On applying the key renewal theorem to (4), (5), (11) and (12), we see that
lim Ait)
cf lim E{Jit)}
Ml +M2
M , + y E i A )/* 2
Ml +M2
Expressions for other availability measures are readily derived but, in general, we do not
obtain formulae which, like those for Akit), are simple modifications of the corresponding
expressions for the alternating renewal model.
236 L A BAXTER
ACKNOWLEDGMENTS
I am indebted to Mr. R. F. Galbraith for several helpful discussions during the preparation
of this paper. I would also like to thank Mr. M. A. Baxter for some useful comments and the
referee for a number of suggestions which improved the presentation.
REFERENCES
[1] Barlow, R.E. and F. Proschan, Statistical Theory of Reliability and Life Testing (Holt,
Rinehart and Winston, New York, 1975).
[2] Baxter, L.A., "Availability Measures for a Two-State System," Journal of Applied Probabil-
ity 18 (1981) (to appear).
[3] Chiang, C.L. and J. P. Hsu, "An Alternating Renewal Process with an Absorbing State" in
Applications of Statistics, 109-121, Editor, P.R. Krishnaiah (North Holland, Amsterdam,
1977).
[4] Cleroux, R. and D.J. McConalogue, "A Numerical Algorithm for Recursively-Defined
Convolution Integrals Involving Distribution Functions," Management Science 22,
1138-1146 (1976).
[5] McConalogue, D.J., "Convolution Integrals Involving Probability Distribution Functions"
(Algorithm 102), Computer Journal 21, 270-272 (1978).
[6] McConalogue, D.J., "Numerical Treatment of Convolution Integrals Involving Distribu-
tions with Densities having Singularities at the Origin," Communications in Statistics, B
(1981) (to appear).
AN ANALYSIS OF SINGLE ITEM INVENTORY
SYSTEMS WITH RETURNS*
John A. Muckstadt and Michael H. Isaac
Cornell University
Ithaca, New York
ABSTRACT
Inventory systems with returns are systems in which there are units re-
turned in a repairable state, as well as demands for units in a serviceable state,
where the return and demand processes are independent. We begin by exa-
mining the control of a single item at a single location in which the stationary
return rate is less than the stationary demand rate. This necessitates an occa-
sional procurement of units from an outside source. We present a cost model
of this system, which we assume is managed under a continuous review pro-
curement policy, and develop a solution method for finding the policy parame-
ter values. The key to the analysis is the use of a normally distributed random
variable to approximate the steady-state distribution of net inventory.
Next, we study a single item, two echelon system in which a warehouse
(the upper echelon) supports N(N > 1) retailers (the lower echelon). In this
case, customers return units in a repairable state as well as demand units in a
serviceable state at the retailer level only. We assume the constant system re-
turn rate is less than the constant system demand rate so that a procurement is
required at certain times from an outside supplier. We develop a cost model of
this two echelon system assuming that each location follows a continuous re-
view procurement policy. We also present an algorithm for finding the policy
parameter values at each location that is based on the method used to solve the
single location problem.
1. INTRODUCTION
Many models have been developed during the past 15 years pertaining to various aspects
of managing repairable item inventory systems (e.g., [1], [4], [10], [11], [12], [15], and [16]). '
Most of these models contain the assumption that the failure of a unit simultaneously generates
a demand for a unit of exactly the same type, i.e., the demand process for serviceable units and
the return processes of failed units are perfectly correlated.
In certain instances, however, this assumption of perfect correlation between the demand
and return processes is not valid. For example, this can occur in situations where equipment is
leased, rented, and/or sold, such as found in the telephone, computer and copying machine
industries. Returns do not necessarily correspond to failures in these cases, but rather to lease
or rental expirations. At the time a unit is returned, it may have to go through a repair or
*This research was supported in part by the Office of Naval Research under Contract N00014-75-C-1 172.
A repairable item is an item which fails, but which can be repaired and subsequently made available to satisfy a future
demand or an existing backorder.
237
238 J. A. MUCKSTADT AND M. H. ISAAC
overhaul process before reissue. There is no reason to assume that the customer will request a
unit of exactly the same type when a lease or rental agreement expires. Similarly, when a cus-
tomer requests a particular type of unit, there is no reason to assume that the customer will
return one of exactly the same type.
The authors studied a real two echelon inventory repair system managed by a manufac-
turer of reprographic equipment. This system closely resembles the one described in Section 3.
For that system we found the demand and return processes to be independent Poisson
processes. That is, we tested and could not reject the hypotheses that the demand and return
random variables had Poisson distributions, and that the return and demand random variables
were independent. The research described in this paper reflects our study of this system's
behavior. Consequently, we assume in the remainder of this paper that the demand and return
processes are independent. We call such inventory systems, inventory systems with returns.
Only a few papers have been published on inventory systems with returns. These papers
contain simplifying assumptions which make them of limited practical value. Heyman [6,7]
considers optimal disposal policies for a single item inventory system with returns; but his
assumptions include instantaneous outside procurement (implying no backorders or lost sales)
and no fixed cost of ordering (implying no lot size reordering). Hoadley and Heyman [8] con-
sider a two echelon inventory system with outside procurement, returns, disposals, and trans-
shipment; but their model is a one period model, and all of the mentioned transactions are
assumed to occur instantaneously. Simpson [16] develops the optimal solution for a finite hor-
izon, periodic review model. His model allows for correlation between the return and demand
processes. Backlogging is permitted, but both repairs and outside procurements are assumed to
be instantaneous.
For the most part, the methods of analysis in these three papers rely heavily upon the
assumptions of instantaneous repair and procurement. Their approaches are of little use when
analyzing situations in which repair and procurement times are not zero.
Finally, Schrady [14] solves for repair carcass and procurement lot sizes for a completely
deterministic system. Gajdalo [2] extends this to a 'continuous review repair policy' for an
inventory system with stochastic (compound Poisson) returns and demands. He uses computer
simulation to test several heuristics for computing the reorder point and lot sizes for both pro-
curing and repairing items. All lead times, including repair times, are assumed constant.
Our approach differs substantially from those taken in these previous studies. We begin
in the next section by analyzing a single item, single location inventory system with returns.
We develop the stationary distribution of two key random variables that describe the probabilis-
tic behavior of the inventory system. This analysis is used as the basis for a cost model. A
solution method is then presented for finding the values of the policy parameters. The results
of the single echelon case are then extended in Section 3 to a specific two echelon situation,
which corresponds to the real environment mentioned earlier. In Section 4, we conclude with a
brief summary and some final comments.
2. THE SINGLE ECHELON CASE
The system we study in this section consists of a single type of item managed at a single
location. A schematic representation of the system's operation is given by Figure 1. As
shown, this location is assumed to contain both a repair facility for returned units and a ware-
house, or storage facility, for serviceable inventory.
SINGLE ITEM INVENTORY WITH RETURNS
239
procurement
source
returns at
rate y
^XXX
D
D
■>
V
serviceable
inventory
repair facility storage facility
demands at
rate X
Figure 1. A schematic representation of the inventory system.
We assume returns of repairable units occur as a Poisson process with rate y, and
demands for serviceable units occur as a Poisson process with rate A.. As we have stated, we
also assume that these two processes are independent, y is assumed to be less than X, so that
an occasional procurement of units from an outside source is required. Units procured in this
manner arrive in a serviceable state r time units after they are ordered.
The repair facility behaves as a first-come, first-served queueing system with Poisson
arrivals (the Poisson returns). All returned units require repair, and repair times of returned
units are independent. Since y < X , the repair system is always operating as long as repairables
are present. No other assumptions about the queueing repair system (e.g., service time distri-
bution or the number of repair servers) are made.
The output of this queueing repair system is input to the stock of on-hand. serviceable
inventory, as is the arrival of outside procurement orders.
All demands that are not satisfied immediately are assumed to be backordered.
We define 'net inventory' at a point in time to be the number of on-hand serviceable
units in the storage facility minus the number of outstanding backorders. We also define
'inventory position' at a point in time to be the sum of net inventory, the number of units in
the repair queueing system, and the number of units on order from the outside procurement
source.
and
Let
lit)
Nit)
Rit)
Pit)
Oit)
Bit)
the inventory position at time r,
the net inventory at time t,
the number of units in the repair queueing system at time ?,
the number of units on order from the outside supplier at time /,
the on-hand serviceable inventory at time t,
the number of outstanding backorders at time /.
240 J. A. MUCKSTADT AND M. H. ISAAC
Then
and
1U)= Nit) + R(t) + Pit),
Nit) = OU) - Bit).
Our final assumption concerns the form of the procurement policy. We assume that a
continuous review (Q,r) procurement policy is followed, i.e., when the inventory position drops
below r + 1, and order for Q ^ 1 units is placed immediately. Since the repair queueing sys-
tem is assumed to be operating continuously, our objective is simply to find values of Q and r.
Our analysis begins with the derivation of the steady-state distribution of inventory posi-
tion. This result is used in the derivation of an approximation of the steady-state distribution
of net inventory, and is followed by a discussion of the accuracy of the approximation.
2.1 Derivation of the Stationary Distribution of Inventory Position
Changes in the state of the inventory position are caused only by demands and returns.
State /(/' = r + \,r + 2, ...) can be entered from state i + 1 when a demand for a serviceable
item occurs; state j (j ' = r + 2,r + 3, . . .) can be entered from state j - 1 when an item is
returned. In addition, state r + Q can also be reached from state r + 1 when a serviceable
item is demanded (an order for Q units is placed immediately when the inventory position
drops below r + 1). The time between state transitions is exponentially distributed, since the
return and demand processes are Poisson processes. The state transition flow diagram is given
in Figure 2, with the transition rates as indicated.
Let u, = lim Prob(/(/) = r + 1 + /), the stationary probability that inventory position is
equal to r + 1 + /. This limit exists because the states of this system are the states of an
irreducible, ergodic, Markov chain [13]. The steady-state balance equations corresponding to
this system are
(1) (A+yh/0= +kuu
(X. + y)u\ = yu0+ \i/2,
(\ + y )ih = yf/]+ \u\.
(X + y)iiQ-\ = yWy-2 + ^uq + ^wo>
(A + y)uQ = yuQ-i + kuQ+l,
SINGLE ITEM INVENTORY WITH RETURNS
A
241
r+1 Y r+2 y r+Q-1 r+Q r+Q+1
FicjURE 2. State transition flow diagram for inventory position.
(2)
A generating function approach can be used to solve for the u,. Define the generating
$(
zQ)
function G(z) to be G(z) = £ z'w,. Using (1) we find that
;=0
G(z) =
X
<? (l-z)(X-yz)'
from which we find that the u, are given by
0 / < 0,
(3)
w,=
1
B-l
(3
i-Q+l
i-U
0 ^ / < Q - 1,
21
/ > a
and the mean and variance of the stationary distribution of inventory position are given by
(4) 7-
and
£[lim lit)} = r + 1 + CO) = r + 1 + ^ * +
f— oo 2
X-y
(5)
respectively.
Varllim Ht)] = G"(l) + G'(l) - [G'(\)]2
Q2-l
12
+
Ar
(X-y)
2 '
If Q = 1, Figure 2 is the transition flow diagram for an Ml MIX queueing system in which
the 'arrival' rate is y and the 'service' rate is X. In this case (3) reduces to the geometric distri-
bution, which is the steady-state distribution of the number of customers present in an Ml MIX
system.
Note that when y = 0, (4), (5), and (3) reduce to the mean, variance, and probability
distribution, respectively, of a uniformly distributed random variable, which is a well known
result (see Reference 5).
2.2 An Approximation to the Stationary Distribution for Net Inventory
Next, we develop an approximation to the stationary distribution of net inventory, which
is the basis for the cost model used to determine optimal values of Q and /•.
242 J. A. MUCKSTADT AND M. H. ISAAC
Recall that t, the procurement lead time, is constant. Thus, any units on order at time
t — r will have arrived by time t. Similarly, any order placed after time t — r will not have
arrived by time /. Therefore, we see that
(6) Nit) = IU-t) - R(t-r) + Z(t - T,t) - D(t - r,r),
where R(t — r) = the number of units in the repair system at time t — r,
Z(t — t ,t) = the output of the repair system in the interval (t — r,t],
and D(t — r,t) = the number of demands in the interval (t — r,t\.
R(t — t) is subtracted from lit — t) so that we do not double count the units in the repair
system at time / — r that complete service by time /. Therefore, net inventory at time t con-
sists of units on order, already serviceable, or backordered at time t — r (all measured in
I(t — r)), plus those units completing repair by time t — t, minus demands over the interval
U-tj].
Let us separately examine the individual terms of (6). The steady-state distribution of
lit — t) has already been obtained. The number of demands over the interval it — r ,t] is
Poisson distributed with mean yr and is independent of the other three random variables on
the right-hand side of Equation (6).
The distributions of R (t — r) and Z(t — t ,t) are readily available for many queueing sys-
tems; but, they are not independent of each other or of /(/ — r). The joint distribution of
these random variables is difficult to develop analytically. Consequently, an approximation to
the distribution of net inventory will be developed, using (6), rather than developing the exact
distribution.
We initially observed that the steady-state distribution of net inventory for numerous test
cases (obtained via simulation) closely resembled a normal distribution. As a result, the nor-
mal distribution was considered to be a candidate approximation to the steady-state distribution
of net inventory.
Equation (6) is used to determine the mean /x and to approximate the variance, o-2, of
this normal approximation. Letting t — - °°, we have
(7) fi = E(NU)) = £(/(/ - t)) - E(RU - r)) + E{Z{t - r,t)) - E(D(t - r,/))
= r + 1+ Q~ 1 + — l E(R(t -t)) +yr -At,
2 k — y
using (5), and noting that the expected output of a queueing system over an interval is equal to
the expected input over an interval of the same length. Also, by ignoring covariance terms, we
approximate a2 by
(8) o-2= Var(/V(/)) = Var(/(r - r)) + Var(/?(r- r))
+ Var(Z(r - T,t)) + Var(D(t - r,t))
= Q ~ l + , Xy „ + VariRU - r)) + Var(Z(f - r,t)) + At,
12 (A - y)2
using (5). Note that exact expressions and good approximations for E(R(t — t)),
V-ariR (t — t)), and Var(Z(? - tj)) are available for many queueing systems (e.g., see [3]).
SINGLE ITEM INVENTORY WITH RETURNS 243
The accuracy of the normal approximation, whose mean and variance are given by (7)
and (8), was tested using an incomplete factorial experiment. The variable factors were the
number of repair servers, the repair service distribution, the repair system traffic intensity, the
procurement lead time r, the procurement lot size Q, and the ratio y/X. In each test case, the
accuracy of the normal approximation was first measured by finding the area between the nor-
mal curve and the curve representing the continuous version of the distribution of net inven-
tory, which was obtained via simulation.
The conclusion drawn from this experiment was that the major factor affecting the accu-
racy of the normal approximation is the ratio of the return rate to the demand rate, y/X. In
fact, the normal approximation is quite accurate when y/X < .6. However, a closer analysis of
the normal curves revealed that the normal approximation was an excellent one in the left-hand
tail of the distribution of net inventory in all the test cases. (We discuss in Section 2.3 why the
left-hand tail of the distribution is all that is needed to determine optimal values for Q and r.)
The difference between left-hand tail percentiles of the experimental distributions and the
corresponding approximating normal distributions were computed. The percentiles never
differed by more than a few percent. Based on this observation we conclude that the steady-
state distribution of net inventory can be accurately approximated by a normal distribution
whose mean and variance are given by (7) and (8), respectively.
2.3 Cost Model and Solution Method
The optimization model we will construct includes a fixed procurement order cost, a hold-
ing cost, and a time-weighted backorder cost. In particular, let
A = the fixed procurement order cost ($/procurement order),
h = the holding cost ($/unit-year),
and n = the backorder cost ($/unit-year).
Our objective function, K, is the sum of the expected annual procurement ordering, holding,
and backorder costs. It will be evaluated by taking the sum of
(1) A x (the expected number of orders placed per year),
(2) h x (the expected serviceable on-hand inventory at a random point in time),
and (3) 77- x (the expected number of outstanding backorders at a random point in time).
Both the expected on-hand inventory and expected backorders at a random point in time will be
calculated using the normal approximation to the distribution of net inventory.
Note that we need not consider holding costs charged against units in repair. Due to the
assumption that no inserted idleness in the queueing repair system is allowed, these holding
costs are independent of the values of the procurement policy parameters.
Let (/>(•) and <£(•) be the standard normal density and standard normal distribution func-
tions, respectively. Let h (x) be the normal density, which is the continuous approximation to
the steady-state distribution of net inventory, whose mean /j, and variance cr2 are given by (7)
and (8), respectively. Thus, the expected number of backorders at any point in time is
(9) o-0
cr
fJL<&
_ IL
cr
244
J. A. MUCKSTADT AND M. H. ISAAC
which can easily be obtained by evaluating
f °
xh (x)dx.
J v=-oo
Since
£ (on-hand inventory) = E (inventory position) + £(backorders)
— £(number in repair) — £(number on order),
the expected on-hand inventory is equal to
(10)
r + 1 + ^ „ l + — * — + o-4>
2 X - y
I \
■
JL
CT
- n<P
_ JL
a
E(RU))- (X-y)r.
Note that the last term, the expected amount on order at any point in time, is equal to the rate
at which demands are ultimately met by outside procurement, X — y, times the constant pro-
curement lead time, r.
In what follows, it will be easier to think of /jl and a2 as functions of /• and Q.
Specifically, let
(11)
fi = r + & + c,
and
(12)
*>=£ + *
where
(13)
K - y 2
and
(14)
„--^-
+ -tr- i:iR(t)) - (X -y)r
12
+ Var(/?(f)) + (X +y)r,
(X -y)2
where we have used the approximation that Var(Z(/ — t, t)) = yr . This approximation is
exact for M\M\s and M\G\°° queueing systems. Note also that the constants c and d are
independent of /• and Q, and that the restriction that Q be greater than or equal to one guaran-
tees that cr2 is positive.
Finally, the rate at which demands are met by outside procurement, X — y, divided by Q,
the procurement lot size, gives the expected number of procurement orders placed per year.
Combining our previous results, we see that the optimization problem for finding the
optimal Q and /• is
(15)
minimize K = — ^ V Or + h)
(T(f)
IT
~ /X<I>
cr
+ h
r + JQ + c
where c is given by (13). This formulation of the problem came as a result of a number of key
assumptions and approximations, which we now summarize:
SINGLE ITEM INVENTORY WITH RETURNS 245
(a) The demand and return processes are independent Poisson processes.
(b) The return rate is less than the demand rate.
(c) A continuous review (Q,r) policy is followed.
(d) The procurement lead time in constant.
(e) All demand not immediately satisfied is backordered.
(f) The distribution of net inventory is approximated by a normal distribution whose
mean and variance are given by (7) and (8), respectively.
The objective function K is not convex in 0, but is convex in r. This is easily proven by
noting that the backorder function o-0
a-
— fj.®
JL
r is not related to a. Thus, the optimal value of r satisfies -r— = 0, that is,
9/'
is convex in /x, r = fx — Q/2 — c, and
BK
(16)
$
_ iL
ar
h
7T + h
Thus, for a fixed value of £), the variance of the normal distribution representing net inventory
is fixed. Only the mean, or "location" of the curve, is decided by choosing a value of r. There-
fore, Equation (16) indicates that once the variance is fixed, the "location" of the normal curve
should be chosen so that the cumulative area to the left of the .y-axis is — — , as illustrated in
Figure 3.
TT + H
shaded area = -
>:<
Figure 3. Location of the normal curve.
In most real situations, the backorder cost v is large compared to the holding cost /?. This
h
makes the fraction small. Recall that this fraction is the area to the left of the v-axis
TT + h
under the normal curve. The expected number of backorders is calculated using Equation (9),
and the expected on-hand inventory is calculated in Equation (10) also using (9). Thus, as we
stated earlier, accuracy of the normal approximation is required only in the tail of the distribu-
h
tion, since is usually small.
TT + h
246
J. A. MUCKSTADT AND M. H. ISAAC
Returning to (16) and rewriting it in terms of r* and Q*, the optimal values of r and Q,
respectively, we have
cp
0*
2
1
+ c
V-
(Q*)2
12
+ d
77
TT + /?
or
(17)
(Q*)2
= V^
TT
-\q*
TT + h
For a fixed value of Q, the optimal value of r is given by Equation (17).
To find the optimal value of Q, one can rewrite Equation (15) in terms of r and Q. Using
Equation (17) to write the objective function solely as a function of Q, (15) simplifies to
K
= (k-y)A + or + hu/£- + d ■ J*-
q V 12
TT
TT + h
This can be seen to be a convex function of Q. While the original objective function, K, is not
convex everywhere in both Q and r, upon deriving an optimality condition (17), AT is convex in
both Q and r over the region of interest. Setting —7- = 0, we find that Q* is the value of Q
that satisfies
(19)
where
V 12
+ d
\2(\-y)A
a
a = (fr + h)(f>
If Q* < 1, then set Q* = \.
<I>
TT
TT
+ h
Note that in realistic situations d > 0 (see Equation (14)), so the left side of (19) should
increase with Q. A search method, such as either the Fibonacci or binary search technique, can
be used to find Q* in this case. Note, also, the similarity to the usual lot size formula. Ignor-
ing some of the constants, (19) is roughly of the form
V(\ - y)A
h
Q — ^
Also, observe that (19) is independent of r. Thus, once Q* is found, r* is found using (17).
3. THE MULTI-ECHELON CASE
In this section we study a two echelon system, which corresponds to the real system
examined by the authors. The upper echelon consists of a warehouse having both repair and
storage facilities that support the /V lower echelon retailers. The retailers only have storage
facilities.
SINGLE ITEM INVENTORY WITH RETURNS 247
All primary customer demands and returns are assumed to occur only at the retailers. We
again assume that all customer demands not immediately satisfied are backordered, and that the
demand and return processes are mutually independent Poisson processes. We also assume
that lateral resupply is not allowed between retailers.
Let
Kj = the customer demand rate at retailer 7(7 =1, . . . , AO,
y, = the customer return rate at retailer j(J = 1, .... N),
T\ = the constant transportation time between the warehouse and a retailer, and
T2 = the constant procurement lead time for the warehouse from an outside source.
The assumptions that transportation times are identical between the warehouse and any of
the retailers, and that customer demands and returns occur only at the retailers are made for
notational simplicity only. It will be apparent that relaxing these assumptions poses no addi-
tional problems.
Recall that repair facilities exist only at the upper echelon. Consequently, we assume that
when a customer returns a repairable unit to a retailer it is immediately sent to the warehouse
from the retailer and need not go back to that same retailer after it is repaired. We also assume
that the repair process at the warehouse operates as a first-come, first-served queueing system.
Since transportation times are assumed to be constant, returns of repairable units to the
N
warehouse occur as a Poisson process with rate y0 = £ "Yj- Therefore, it is equivalent, and
j=\
more convenient, to think of returns occurring only to the warehouse, and as a Poisson process
with rate y0.
We assume that retailer j uses an (Sj — 1, Sj) continuous review ordering policy, i.e., a
constant inventory position (net inventory plus on order) of Sj is maintained. This implies that
retailer j immediately orders one unit from the warehouse every time a customer demand
occurs at the retailer. Since each order placed at a retailer also results in a demand being placed
N
upon the warehouse, demands on the warehouse occur as a Poisson process with rate \0 = X
\j.
[Note the importance of the assumption of following an (Sj — 1, Sj) policy at retailer /
If the retailers followed (Q,r) ordering policies, then the time between the placing of orders
upon the warehouse would not necessarily be exponential, nor would the orders necessarily be
for individual units. Thus, the demand process at the warehouse would no longer be a simple
Poisson process.]
We assume that y0 < ^0 so that an occasional outside procurement is necessary. The
warehouse is assumed to follow a (Qo-^o) policy, i.e., when its inventory position (net inven-
tory plus on order plus in repair) falls below r0 + 1 , an order for Q0 units is placed upon an
outside procurement source.
248
J. A. MUCKSTADT AND M. H. ISAAC
Warehouse procurement orders are assumed to arrive at the warehouse T2 time units after
the order is placed. However, an order placed by a retailer upon the warehouse does not neces-
sarily arrive at the retailer Tx time units after it is placed. In addition to the transportation
time, there may be a delay due to the warehouse being out of serviceable stock. All demands
made upon the warehouse that are not immediately satisfied are backordered.
A schematic representation of this system is given by Figure 4.
Warehouse
Return rate Yn
Demand rate A
Lead time T
(Qo'ro) p°licy
V
Retailer j
Demand rate A.
]
Transportation time T.
(Sj - 1, Sj) policy
FlGUR] 4. Schematic representation of the multi-echelon system.
Finally, let the system cost parameters be as follows:
h0 = the holding cost at the warehouse ($/unit— year),
//; = the holding cost at retailer ./($/ unit-year) (J = 1, .... A/),
tt j = the backorder cost at retailer y($/unit- year) (j = 1 A/),
and A = the fixed warehouse procurement order cost ($/order).
Given values of hjij = 0, . . . , A/), -n-jij = 1, . . . , A/), and A, all assumed to be nonne-
gative, the problem is to determine values for (?rji fo> and Sj(j = 1, .... N) that will minimize
the expected annual sum of the retailer holding and backorder costs, and the warehouse order-
ing and holding costs. Thus, the optimization problem we want to solve is
(20) min
Qo-ro-sj
SINGLE ITEM INVENTORY WITH RETURNS 249
N
£ (hj ■ £ {On-hand Inventory at Retailer j)
+ ttj • E {Backorders Outstanding at a Random
Point in Time at Retailer j))
+ A ■ — — 1- /?o • £ {On-hand Inventory at the Warehouse} |
Qo j
subject to Q0 > 1, /-0 > 0 and 5,- = 0, 1, .... for j = 1, .... N.
The expected on-hand inventory at the warehouse can be found using Equation (10); however,
the expected on-hand inventory and backorders at retailer j cannot be determined as easily. We
will subsequently show how these expectations can be calculated.
Note that we have not explicitly stated a value for fr0, the warehouse backorder cost, and
that this cost is not included in the objective function that is to be minimized. Given the
interactions between the two echelons of our inventory system, the cost of a backorder at the
warehouse is not an explicit one but rather an imputed one. It is measured by the effect of a
backorder at the upper echelon on the expected performance at the lower echelon.
The optimal stock level at retailer j, SJ, is a function of the procurement resupply time,
that is, the expected time from the placement to receipt of an order by a retailer. This procure-
ment resupply time is then the transportation time, T\, plus the expected delay due to the
warehouse being out of serviceable stock. Clearly, costs at the retail echelon can be lowered by
reducing the expected resupply time. This can only be accomplished by decreasing the
expected warehouse backorders at a random point in time, which is achieved by increasing Q0
or rQ (or both). This, in turn, raises holding costs at the warehouse. Thus, a tradeoff exists
between holding costs at the upper echelon and holding and backorder costs at the lower
echelon. We will present an iterative algorithm based on this tradeoff which alternates between
finding stock levels for the upper and lower echelons. The basis for this algorithm, presented
in Section 3.1, is founded on the results developed in Section 2.
3.1 Analysis
Suppose the imputed cost of a warehouse backorder is known to be fr0. Then we can use
(17) and (19) to find optimal values for r0 and Q0. These determine a "performance level" B,
where
B = the expected backorders at the warehouse at a random point in time
1
cr
- /u<£>
cr
and where /x and cr2 are the mean and variance, respectively, of the normal approximation to
the stationary distribution of net inventory at the warehouse.
Then the expected resupply time for a retailer is
(21) T= Ti + BlkQ,
250 J. A. MUCKSTADT AND M. H. ISAAC
since the expected delay time per demand is the expected number of backorders at a random
point in time divided by the demand rate. This is a direct application of Little's Formula
L = \ W. Then, using Palm's Theorem [1] as an approximation, we assume the number of
units in resupply at retailer j to be Poisson distributed with mean \j T.
Note: Palm's Theorem requires the independence of resupply times, making this system
analogous to an M/G/°° queue. Resupply times in our system are not independent; consider,
for example, a demand by a retailer which cannot be immediately filled by the warehouse.
Then it is more likely that the next demand placed by a retailer upon the warehouse also
experiences a delay than if the preceding order had been immediately satisfied. This approxi-
mation of the distribution of the number of units in resupply at retailer j(j = 1 A/) was
tested for the special case in which the repair facility at the warehouse behaves as an M/D/°°
queueing system. The exact distribution of Rj(t), the number of units in resupply at retailer y,
was obtained from comparison with the Poisson approximation. Our analysis indicates that the
Poisson approximation improves as the expected warehouse backorders, or the probability of
delay at the warehouse, decreases. In particular, the Poisson approximation was found to be
good as long as the expected value of net inventory at the warehouse at a random point in time
is greater than zero. (In the test cases in which this condition was met the maximum absolute
difference between R/(t) and its Poisson approximation was less than 5%.) This will, of
course, be the case for a reasonably large ratio of backorder to holding costs.
Once we know the value of T and have the form (approximately) of the distribution of
the number of units on order by retailer j, we can solve N independent subproblems to obtain
the optimal value for S,-. The subproblem at retailer j consists of finding the optimal stock level
5*, assuming a constant procurement resupply time of T, where T is given by (21). This is
accomplished using Lemma 1 .
LEMMA 1: Suppose the procurement lead time is a constant T and demand is Poisson
distributed with rate \r Then the optimal value S* for an (S, — 1, Sj) policy is the largest
integer S7 such that
(22)
P( <? ■ \ T) >, J
r \o/ , A. j i ) ^>
TTj + hj
DO
where
^Um) = I P^r-fx)
r=x
and
p(r,fi)= e-» **j.
The proof of Lemma 1 can be found on page 204 of Reference 5.
Let Kj(SJtT) be the expected annual holding and backorder costs at retailer j when the
inventory position is S, and the procurement lead time is a constant T. As can be shown (see
Reference 5)
(23) KjiSj, T) = (it j + hj) [A, TPiSj - 1 ; A, 3D
- SjP(Sj\\jT)] + hj[Sj-\jT].
For a fixed value of T (and therefore of B) we define the minimum total expected costs at the
lower echelon, K'iB), as
(24) K'iB) = £ Kj{S*,T),
SINGLE ITEM INVENTORY WITH RETURNS
where Kj(- , •) is given by (23), Tis given by (21), and S* satisfies (22).
dK'(B)
251
Note that when B = b,-
dB
is an estimate of 7r0, since it measures the marginal
B=b
effect of a warehouse backorder on the expected total lower echelon cost. It is easy to show
that
-Kq\ 1 jy
(25)
^jf- " T- £ K*j + hj^jHSJ; KjT) - hjXjl
dB
L0 j=\
Next, let KU(B) represent the minimum expected warehouse ordering and holding cost
given that B, the expected number of warehouse backorders outstanding at a random point in
time, is fixed. In particular, we define
i
xo~yp ...
— t; • A + h0
KU{B) = min
Q0>\
r0+ — + B + c
rn>0
subject to B = X0(T -\ T\),
where the constant c is given by Equation (13).
We conclude this section with the statement of two additional lemmas.
LEMMA 2: K"(B) is convex decreasing in B.
LEMMA 3: Let T be a constant resupply time. If the optimal stock levels Sj(j =
1, . . . , N) are continuous rather than integer valued, then K'(B) is a concave increasing func-
tion of B, where B = ko(T — T\). These lemmas can be proved by applying the chain rule to
take derivatives. The details can be found in Reference 9.
3.2 Restatement of Problem 20
Problem (20) can be restated based on the interrelationship between the warehouse and
the retailers developed in Section 3.1. As we have demonstrated, the two echelons are linked
through the value of B. Then an alternative way of writing problem (20) is
(26)
where
min K'(B) + KU{B)
B = X0(T- Td.
Figure 5 represents a typical graphing of K'iB) and KU(B) as functions of B. We observed in
all test cases that, under the conditions of Lemma 3, KU{B) + K'iB) was a convex function of
B. Thus, the minimum cost will occur where
(27)
dK'(B)
dB
dKHB)
dB
The algorithm presented in the next section takes advantage of the fact that problem (20) can
be restated as problem (26) and that the optimal solution must satisfy (27).
252
J. A. MUCKSTADT AND M. H. ISAAC
cost
> 3
Figure 5. Minimum upper and lower echelon cost functions vs. B.
3.3 An Algorithm
The following algorithm can be used to solve problem (26):
STEP 0: Let 77-n = max (77,).
j=\ jv J
STEP 1: Given 7r0, solve for Q0J0 using Equations (17) and (19) and determine the
corresponding value of /?, say b.
STEP 2: Let T = Tx + b/k0\ find the S* using Equation (22).
dK'(B)
STEP 3: Using these SJ, find
dB
evaluated at B = b, using Equation (25); let n0 assume
this value, and return to Step 1 unless the stock levels and costs have converged
sufficiently.
The first few steps of the above algorithm are illustrated in Figure 6. The algorithm
begins by setting n0 = max (77-,). This is an upper bound on the optimal value of 7r0, since
this value implies that a backorder at the warehouse always results in a backorder at the retailer
with the largest backorder cost. Then Q0 and r0 are found using this upper bound on 7r0. This
determines a value of B (say B = b\) (and therefore of 7), which is a lower bound on the
optimal value of B (and therefore of 7). These computations yield point (T) on the upper
echelon cost curve in Figure 6.
Using this lower bound on the optimal value of T, we find a lower bound estimate of
5*(/ = 1 N), which determines a value K'(b\), and point (2) in Figure 6. Next we set
dK'(B)
7T0 =
dB
B=b,
. Since K'(B) is concave in B, and since we have a lower bound estimate
of the optimal B, the new estimate of tt0 is an upper bound on the optimal value of 7r0; but it is
smaller than the previous estimate. Using this new estimate of -fro, B will increase to a value,
say b2, as a result of resolving for r0 and Q0 using (17) and (19). These calculations produce
point (T) in Figure 6. The procedure continues by letting T= T\+ — and finding K'(b2),
which leads to point [4). The algorithm continues in this manner until convergence occurs.
Discussion of convergence and other aspects of the algorithm can be found in Reference 9.
The algorithm was tested on 50 problems. In general, the values of (>*, r* and S*
0=1 AO were found after only three iterations of the algorithm. This occurred in 48 of
the 50 test cases. The curve K'(B) is very flat compared to K"(B), so that convergence to the
SINGLE ITEM INVENTORY WITH RETURNS
253
cost
Figure 6. First steps of the algorithm
correct value of no occurs quickly. As we noted earlier, K'(B) + K"(B) was convex for all of
the 50 test problems. The reason this occurred was because K'(B), although concave, is
almost linear.
4. SUMMARY AND CONCLUDING COMMENTS
We have developed simple methods for obtaining parameter values for a procurement pol-
icy for certain inventory systems with returns. The key was the use of a normal approximation
to the steady-state distribution of net inventory. This led to the development of cost models
which were easily solved.
In the single location model, we assumed the procurement policy to be a stationary (Q,r)
policy. This policy is not the optimal one. In Reference 9 it is shown that, for the special cases
of Ml MIX and M/G/°o queueing repair systems for which the transient distributions of the
repair system's output are easily developed, one can lower total expected costs by redefining
inventory position and allowing variable reorder points as follows. Inventory position is
redefined to be net inventory plus the number of units on order. The analysis proceeds exactly
as described in Section 2 (with some of the constants redefined). This results in a reduction in
cr2, the variance of net inventory, since the variance of the number of units in repair is no
longer included in a2. The reorder point, expressed in terms of inventory position, is then a
function of the number of units in repair, rather than a constant. Reductions in total expected
costs can be achieved by using a state dependent reorder point when the variance of the
number of units in repair is very large. A 10% reduction in total expected cost was achieved
using the variable reorder point policy in an Ml MIX repair system with traffic intensity
p = 499/500. This is an extreme case, however. The average annual cost of using the station-
ary (Q,r) policy was within 1% of the average annual cost obtained using the nonstationary one
in almost all test cases. Since this is the case, and since a stationary (Q,r) policy is easy to use,
the stationary (Q,r) policy is an attractive policy to implement.
Next, we showed how the single location solution method can be incorporated into an
iterative algorithm for setting stock levels in the single item, multi-echelon inventory problem
with returns. The algorithm can also be extended to find stock levels in an M-echelon inven-
tory system with returns. The only requirement would be that an (S — X, S) procurement pol-
icy must be followed at each of the lower M — X echelon locations.
254 J. A. MUCKSTADT AND M. H. ISAAC
REFERENCES
[1] Feeney, G.J. and C.C. Sherbrooke, "The (S — 1, 5) Inventory Policy Under Compound
Poisson Demand," Management Science, 12, 391-411 (1966).
[2] Gajdalo, S., "Heuristics for Computing Variable Safety Levels/Economic Order Quantities
for Repairable Items," AMC Inventory Research Office, Institute of Logistics Research,
U.S. Army Logistics Management Center, Fort Lee, VA (1973).
[3] Gross, D. and CM. Harris, Fundamentals of Queueing Theory, (John Wiley and Sons, New
York, 1974).
[4] Gross, D., H.D. Kahn and J.D. Marsh, "Queueing Models for Spares Provisioning," Naval
Research Logistics Quarterly, 24, 521-536 (1977).
[5] Hadley, G. and T.M. Whitin, Analysis of Inventory Systems, (Prentice-Hall, New Jersey,
1963).
[6] Heyman, D.P, "Optimal Disposal Policies for a Single-Item Inventory System with
Returns," Naval Research Logistics Quarterly, 24, 385-405 (1977).
[7] Heyman, D.P., "Return Policies for an Inventory System with Positive and Negative
Demands," Naval Research Logistics Quarterly, 25, 581-596 (1978).
[8] Hoadley, B. and D.P. Heyman, "A Two-Echelon Inventory Model with Purchases, Disposi-
tion, Shipments, Returns, and Transshipments," Naval Research Logistics Quarterly,
24, 1-19 (1977).
[9] Isaac, M.H., "An Analysis of Inventory Systems with Returns," unpublished Ph.D. disser-
tation, School of Operations Research and Industrial Engineering, Cornell University
(1979).
[10] Miller, B.L., "Dispatching from Depot Repair in a Recoverable Item Inventory System: On
the Optimality of a Heuristic Rule," Management Science, 21, 316-325 (1974).
[11] Muckstadt, J. A., "A Model for a Multi-Item, Multi-Echelon, Multi-Indenture Inventory
System," Management Science, 20, 472-481 (1973).
[12] Porteus, E.L. and Z. Lansdowne, "Optimal Design of a Multi-Item, Multi-Location, Multi-
Repair Type Repair and Supply System," Naval Research Logistics Quarterly, 21, 213-
237 (1974).
[13] Ross, S.M., Introduction to Probability Models, (Academic Press, New York, 1972).
[14] Schrady, D.A., "A Deterministic Inventory Model for Repairable Items," Naval Research
Logistics Quarterly, 14, 391-398 (1967).
[15] Sherbrooke, C.C, "METRIC: A Multi-Echelon Technique for Recoverable Item Control,"
Operations Research, 16, 122-141 (1968).
[16] Simpson, V.P., "Optimum Solution Structure for a Repairable Inventory Problem," Opera-
tions Research, 26, 270-281 (1978).
ANALYTIC APPROXIMATIONS FOR (s,S) INVENTORY
POLICY OPERATING CHARACTERISTICS*
Richard Ehrhardt
Curriculum in Operations Research and Systems Analysis
The University of North Carolina at Chapel Hill
Chapel Hill, North Carolina
ABSTRACT
The operating characteristics of (s,S) inventory systems are often difficult to
compute, making systems analysis a tedious and often expensive undertaking.
Approximate expressions for operating characteristics are presented with a view
towards simplified analysis of systems behavior.
The operating characteristics under consideration are the expected values
of: total cost per period, period-end inventory, period-end stockout quantity,
replenishment cost per period, and backlog frequency. The approximations are
obtained by a two step procedure. First, exact expressions for the operating
characteristics are approximated by simplified functions. Then the approxima-
tions are used to design regression models which are fitted to the operating
chracteristics of a large number of inventory items with diverse parameter set-
tings. Accuracy to within a few percent of actual values is typical for most of
the approximations.
1. INTRODUCTION
There are many situations in which an inventory system's designer can use estimates of
operating characteristics of the system. For example, management may desire forecasts of
inventory on hand, or system operating costs. Our goal in this paper is to develop simple
approximations that designers can use to estimate the following operating characteristics of a
periodic-review inventory system: average holding cost per period, average backlog cost per
period, frequency of periods without backlogs, average replenishment cost per period, and aver-
age total cost per period. These characteristics are defined mathematically in Section 2.
We consider a periodic-review, single-item inventory system where backlogging is permit-
ted and there is a fixed lead time between placement and delivery of an order. Demands during
review periods are represented by independent, identically distributed random variables having
mean /jl and variance a2. Replenishment costs are composed of a setup cost K and a unit cost
c. There is a fixed lead time L between the placement and delivery of each replenishment
order. At the end of each review period, a cost h or p is incurred per unit on hand or back-
logged, respectively. The criterion of optimality is minimization of the expected undiscounted
cost per period over an infinite horizon.
Under these assumptions it has been shown that there exists an optimal policy of the
(s,S) form (Iglehart [3]). That is, a replenishment order is not placed unless the inventory
position (on-hand plus on-order minus backorders), x, is less than or equal to s, at which time
*This research was supported by contracts with the Office of Naval Research and the U.S. Army Research Office.
255
256 R EHRHARDT
an order of size S — x is placed. Computational methods have been developed (Veinott and
Wagner [6]) for calculating optimal policies and their operating characteristics. Unfortunately,
the computational effort required is prohibitive for practical implementation. Furthermore,
exact computation requires the complete specification of the demand distribution, a level of
detailed information that is unlikely to be available in practice.
In this paper we develop approximations for operating characteristics in a two step pro-
cedure. We start with exact analytic expressions for the operating characteristics and approxi-
mate the exact expressions with simplified functions. Then we generalize the functions and fit
their parameters to the observed characteristics of 576 items using least-squares regression.
The resulting approximations are accurate and require for demand information only the mean
and variance. In Section 2 we derive the simplified functions from exact expressions for the
operating characteristics, and in Section 3 we present the results of the regression analyses.
Finally, in Sections 4 and 5 we analyze the accuracy of the approximations and draw conclu-
sions.
2. ANALYTIC APPROXIMATIONS
Consider the model of Section 1 and assume that demand follows a probability density
$(•) and cumulative distribution <t>(). Let 4>*"i) and <£*"(•) be the //-fold convolutions of
these functions. We consider the following operating characteristics of fixed, infinite-horizon
(s,S) policies:
(1) H = average holding cost per period,
B = average backlog cost per period,
P = backlog protection, i.e., frequency of periods without backlogs,
R = average replenishment cost per period, and
T = average total cost per period.
Let
m
(■) = £ <b *"(•),
and
Mi) = £ **"(•).
The functions mi-) and Mi) are renewal functions which govern the frequency of replenish-
ments, and, therefore, the evolution of the inventory positions. We have, as in Roberts [4],
the exact relationships
(2) H= h[\ + Mif))\
I) ~S-y
S So }{S~ y~ xH*(L+uix)miy)dxdy
+ jQS iS - x)<t>*(l+u ix)dx\
B = p[Hlh + (L + \)fjL - S] + p[\ + MiD)] ' Jo ymiy)dy
P= [1 + M(D)] ' f %*"■+» iS- y)miy)dy+<&*(L^) iS)
J 0
R = K[\ + MiD)]~]
T= H + B + /?,
(s.S) INVENTORY POLICY OPERATING CHARACTERISTICS 257
where
D = S - s.
Notice that a constant term cp has been omitted from the expression for replenishment cost R
since it does not affect the choice of an optimal policy. It is difficult to obtain any insights from
(2) regarding the sensitivity of the operating characteristics to values of model parameters.
Indeed, it is exceedingly complicated just to calculate values of the characteristics for a given
set of parameter values. We proceed to simplify the form of expression (2) by introducing
approximations for the functions m(-), M(-), and 0*(L+1)(-).
Replenishment frequency in (2) is given by [1 + M(D)]~]. To approximate A/(), we
use the following result of Smith [5]:
Mix) = x/ix + <r2/(2p2) - 1/2 + o(l)( x — oo.
This yields the approximate value for replenishment frequency
(3) [1 + M(D)]-l^fx/[D + (fi + a-2/p)] =p.
To obtain approximations for the other characteristics in (2), we first need to find a sim-
ple expression for the function m(-). We identify the quantity (S - y) in (2) as the inventory
position (after ordering), with stationary distribution function F() given by
F(S-y) =
Miy)/[\ + M(D)] , s < 5 - y < S
1 ,'S-y-S.
The probability density /(•) of the inventory position (after ordering) on the interval [s,S) is
f(S-y) = m(y)/[l + M(D)l
We approximate /(■) by a constant c on the interval [S,s). There are two reasons why this
should be a reasonable approximation. Firstly, the result of Smith [5] shows that miy) is
asymptotically constant as y grows large. Secondly, we know that /(•) is exactly constant for
the special case of an exponential demand distribution.
We find a value for c by normalizing the approximated distribution. Starting with an exact
expression, we have
(5) J*f(S-y)dy = M(D)/[\ + M(D)l
Then we substitute c for /(•) on the left side of (5) and use (3) on the right side of (5), yield-
ing
(6) c=(l-p)/D.
We use (3) and (6) in (2) to get
(7) H
[(1 - p)/D) fo° fQS~y (S-y- x)0*(Z- + ,) (x)dxdy
+ PJoS(S-x)0*(i+1) (x)dx\
B = p[H/h + (L + D/x - S + (1 - p)D/2]
P = p<D*^ + i)(5) + Ul-p)/D] f%*(i + 1) (S-y)dy
j o
R = pK.
258 R. EHRHARDT
The expressions for //, #, and P in (7) still require the specification of the demand distri-
bution. We obtain a further simplification by approximating the demand distribution with a
gamma distribution. As we show below, this approximation leads to expressions for //, 5, and
P that require for demand information only the mean n and variance a2. The class of gamma
distributions provides good fits for a wide variety of unimodal or nonincreasing densities on the
positive real line and should be a reasonable approximation in our application. For inventory
items that have significantly non-gamma demand distributions, an analyst could produce a new
set of approximations by making the appropriate substitution in (7) and proceeding in the
manner described below.
Let g(-\a,B) be a gamma density function with parameters a and/3. Then we have
(8) <t>*iL + u(x) = g(x\a,B)l
xa-'exp(-;c//3)/[r(a)/3u] , x > 0
0 , x < 0
X
<P*<L + \)(x) = (J(x\a 0) = j ^ g(y\at fify
where
a = (L + D/LtVo-2
3 = ar2/fi.
We define the notation
[fix) |*s /(*)-/(«),
and use (8) in (7) to yield
(9) // = ph[SG(S\a,B) - aBG(S\a + 1.0)]
+ [Ml ~p)/2D] {x2G(x\a,3) - 2aBxG(x\a + 1./8)
, 5
+ (a + Da/32 G(x\a + 2,(3) \
B = pW/h + (Z, + 1) iM - S + (1 - p)D/2]
S
P = PG(S\a,p) + [(1 - p)/D)[xG(x\a.p) - a(3G(x\a + 1,0) |
R = pK.
Observe that the approximations (9) depend on the values of s,S, the economic parame-
ters, and the mean and variance of demand. The function G must be calculated by a numerical
procedure. We use a series expansion for G(x \a,B) when x is less than the minimum of 1 and
a/3, and a continued-fraction expansion otherwise. The procedure is part of a package of com-
puter programs entitled "The IMSL Library" which is marketed by the International Mathemati-
cal and Statistical Libraries, Inc., Houston, Texas.
Despite the effort required to compute (7, the expressions in (9) are an enormous
simplification over (2). In Section 5 we mention the possibility of using a normal distribution
function in lieu of the function G Employing the normal distribution would facilitate manual
computations of the approximations we derive below.
3. NUMERICAL ANALYSIS
In this section we use expressions (9) to develop regression models for the operating
characteristics. We fit the parameters of the regression models to the observed characteristics
(s,S) INVENTORY POLICY OPERATING CHARACTERISTICS
259
of 576 items. The 576-item system is formed by using a full factorial combination of the
parameters in Table 1. Discrete demand distributions are used in the analysis with means rang-
ing from 2 to 16 and variances ranging from 2 to 144. Although the expressions in (9) are
based on a continuous demand distribution, we will show that they can be used to approximate
many of the characteristics of items with discrete distributions, which are more common in
practice. Notice that all the items in Table 1 have a unit holding cost /; of 1. Since the total
cost function is linear in K, p, and /?, we have used h as a normalizing parameter.
TABLE 1 — System Parameters
Factor
Levels
Number
of Levels
Demand distribution
Poisson (o-2//x = 1)
Negative Binomial (a-2//x = 3)
Negative Binomial ((t2//jl = 9)
3
Mean demand (jjl)
2,4, 8, 16
4
Replenishment lead time (L)
0, 2,4
3
Replenishment setup cost (AD
32,64
2
Unit penalty cost (p)
4, 9, 24, 99
4
Unit holding cost (h)
1
1
Policy
Optimal policy,
power approximation policy
2
The (s,S) policies used in the 576-item system are of two types: those with optimal values
of 5,S computed with the algorithm of Veinott and Wagner [6] and approximately optimal
values of s,S computed by the power approximation algorithm of Ehrhardt [1]. For each item
in the system we use the methods in [6] to compute exact values of the characteristics in (2)
and use these as data for our regression analyses. The approximations we obtain are labelled
with subscript "a" when they are used for all 576 items. Subscripts "a,p" or "a,o" are used to
label expressions that apply only to power approximation or optimal policies, respectively.
We develop our regression adjusted approximations in the following subsections. In each
subsection, we derive an approximation and assess its accuracy in the 576-item system. The
measure of accuracy we use is the absolute value of the percentage difference between the exact
and approximated values for individual items. We note here that the accuracy of the approxi-
mations appears to be even greater when the operating characteristics are aggregated over por-
tions of the 576-item system. That is, there are essentially no systematic errors with respect to
any of the model parameters. For a more detailed discussion of this point, see [2].
An Approximation for Replenishment Cost
We use (3) in (9) to obtain the expression for replenishment cost
R =fxK/[D + (ai +cr2/n)/2].
We manipulate the expression to form a linear regression model
(41 KIR) = AQ + Ax D + A2fi + A2{(t2Ipl) + e,
260 R. EHRHARDT
where A0, . . . , A3 are constants to be fit and e is the error term. We use least-squares regres-
sion to fit the model to the system of 576 inventory policies in Table 1. That is, for each of
these policies we use D, ll, and o-2/ll as independent variables, and we use the exactly com-
puted value of llK/R as the dependent variable. The result is the following numerical approxi-
mation for R:
(10) Ra = Kfi/W + (fx +a2/fx)/2- .5121].
which has a coefficient of determination (fraction of variance explained) of 0.9999 for the quan-
tity ix K/R.
When used in the 576-item system, expression (10) is within 0.1% of actual values of R,
on the average. The expression is accurate to within 2% for all but 2 items, with a maximum
error of 2.5%.
An Approximation for Holding Cost
We can treat the unit holding cost as a redundant (normalizing) parameter in our model,
and so we divide the holding cost expression in (9) by // yielding
H/h = (ASG(S\a,(3) -a/3G(S\a + 1./3)]
+ [(1 - p)/2D] {x2G(x\a,/3) - 2a/3xG(x\a + 1,0)
+ (a + l)a/32G(x|a + 2,0) |s<
We take advantage of our improved estimate of replenishment frequency from (10) and replace
p with
(11) r = RJK =fx/[D + (ll +<t2/ix)/2- .5121].
The result is a quantity that we denote as W, given by
(12) W = r[SG(S\a,p)-a(3G(S\a + 1.0)]
-I- [(1 - r)/2D] {x2G(x\a,p) - 2afixG(x\a + 1.0)
+ (a + l)a02<7Oc|a + 2.0) I .
We calculated values of W in the 576-item system. We compared them with the actual values
of Hi h and found a systematic variation with respect to /x and ct2/li. This motivates the linear
regression model
H/h = A0+ A\W + A^ + A}((t2/(x) +€.
where A0, . . . , A2 are constants to be fit and € is the error term. We use least-squares regres-
sion to fit the model to the system of 576 items. The result is a coefficient of determination of
0.9999 for the approximation
(13) Ha= h{W- .1512/u + .1684o-2//li + .0689).
Expression (13) is within 0.7% of actual values of H, on the average, when used in the 576-
item system. It is accurate to within 2% for 96% of the items, and within 4% for 99% of the
items. Only one item produces an error in excess of 6%. This error is 9.2% for the item con-
trolled with optimal values of (5,5) , ll equal 2, a-2 equal 18, pi ' h equal 4, K/h equal 32, and L
equal 0. In general, the largest errors occur for high values of variance-to-mean ratio and low
values of other parameters.
(s,S) INVENTORY POLICY OPERATING CHARACTERISTICS 261
An Approximation for Backlog Protection
Backlog protection is defined as the frequency of periods without backlogs, that is, one
minus the backlog frequency. Since it is a critical measure of service, it is of central interest to
the inventory systems designer. Unfortunately, when (9) is used to construct regression
models for backlog protection, very poor fits result. The highest coefficient of determination
obtained using this approach is 0.68.
We revised the regression model to reflect a theoretical result. When demand is continu-
ously distributed, an optimal policy yields (p/h)/(\ + pi h) for backlog protection. When the
demand distributions are discrete, (p/ h)/(l + pi h) is a lower bound on P for optimal policies.
It was observed in [1] that the power approximation and optimal policies differed in their back-
log frequency performance. Therefore, we decided to fit the two policy rules separately.
We use the model
(1 + p/h)P= Ao + A^p/h) +e,
which dramatically improves the fit. For optimal policies, the simple expression
(14) Pao = (0.0857 + p/h)/{\ + plh)
yields a coefficient of determination of 0.99999 for (1 + pi h)P. We have the same coefficient
of determination for power approximation policies with
(15) Pap = (0.0695 + plh)/(\ + plh).
When used in the 576-item system, expressions (14) and (15) are accurate to within 0.7% on
the average. They are accurate to within 2% for 92% of the items and to within 4% for 98% of
the items. All nine items with errors in excess of 4% have power approximation policies with a
unit penalty cost of 4. The approximations are especially accurate for large unit penalty costs.
An Approximation for Total Cost
We obtain an expression for total cost by summing cost components //, B, and R, and
using approximations (9) for B and R
T= H + B + R
= (1 + p/h) H + p[(L + IV - S + (1 - P)D/2] +PK.
We divide by h, replace p with r, as given by (11), and use approximation (12) for //to obtain
(16) T/h = (1 + plh) W + p/h [(L + Dp - S + (\ - r)D/2] + rK/h.
As we discovered in obtaining a fit for holding cost, a group of related terms should be added
to (16) to obtain a good fit to the system's data. The linear regression model we employed is
T/h = A0 + A\ W + A2(Wp/h) + A3[(L + \)pp/h] + A4(Sp/h)
+ A5(Dp/h) + A6(rDp/h) + A7(rK/h) + A%(p/h)
+ A9(rp/h) + A]0[(L + \)p] +AUS+ AnD + An(Dr)
+ A\4r + A\Sp + Axk((T2lp) + Ax-jip.pl h)
+ AU[((T2/p) (p/h)] +€.
262 R. EHRHARDT
We fit the model to the system of 576 items using stepwise least-squares regression. The
following expression yields a coefficient of determination of 0.998:
(17) Ta= 1.110 hW - .001049/7^+ .3364 to-
-.2234/7 + .3274 hD + .4476//o-2/m + .003062/? o-2/^.
Expression (17) is within 1.9% of actual values of T, on the average, when used in the 576-
item system. It is accurate to within 4% for 89% of the items and to within 8% for 99% of the
items. Only four items produce errors in excess of 10%. These items have fx equal 2, cr2 equal
18, L equal 0, and pi h equal 4 or 9.
Although the approximation appears to be accurate in most cases, it may be inaccurate for
policies that have significantly suboptimal values of 5 and S. This is because the differences
between (16) and (17) suggest that the economics of optimal policies are intrinsic to the
approximation obtained. The robustness of (17) is discussed explicitly in Section 4.
Approximating Backlog Cost
Attempts at finding a simple, accurate approximation for backlog cost were unsuccessful.
Expression (9) was used to construct a regression model similar to those described above. The
result was a coefficient of determination of 0.44. The relative errors were very large, in some
cases exceeding 100%, making them significant even when compared on an absolute basis with
other components of total cost.
The next attempt was to employ the identity
B= T - H - R
and use (10), (13), and (17) in place of R, H, and T. This approximation has an average per-
centage error of 18%, with large absolute errors for many of the items.
In order to get a reasonably accurate approximation, it was necessary to form a regression
model that included all the variables appearing in the models for R, H, and T. It was also
necessary to fit this model separately for optimal and power approximation policies and for each
of the four settings of unit backorder penalty cost. That is, the 576-item system was partitioned
into 8 systems of 72 items, and 8 separate regressions analyses were performed. The resulting
approximation has an average coefficient of determination of 0.998. As the high coefficient of
determination indicates, the fits are good in terms of absolute errors, although there are relative
errors in excess of 70% for items with large values of pi h. However, the approximation is a
complicated expression involving ten coefficients in each of the 8 subsystems (80 coefficients in
all, for the 576-item system). Also, since the approximation was fit separately for each setting
of pi h, there is no explicit functional dependence on this parameter. The reader is referred to
[2] for additional details.
Backlog cost has proven to be surprisingly difficult to approximate. We point out that
among the operating characteristics listed in (2), backlog cost is the most sensitive to the tail of
the demand distribution. It appears that an accurate specification of the demand distribution is
required for a reasonably precise calculation of backlog cost.
4. COMPUTATIONAL EXPERIENCE
We test the quality of approximations (10), (13), (14), (15), and (17) by using them in a
multi-item system with the parameter settings of Table 2. Note that all the numerical parame-
ters have values not found in the 576-item system. Each parameter has one interpolated value
(s.S) INVENTORY POLICY OPERATING CHARACTERISTICS
263
TABLE 2 — A 64-Item System with New Parameter Settings
Factor
Levels
Number
of Levels
Demand distribution
Negative Binomial (cr2/(x = 5)
Negative Binomial (o-2//* = 15)
2
Mean demand
0.5, 7.0
2
Replenishment lead time
1, 6
2
Replenishment setup cost
16, 48
2
Unit penalty cost
49, 132
2
Unit holding cost
1
1
Policy
Optimal policy,
power approximation policy
2
and one extrapolated value. A full factorial combination of the values is used, yielding 64
items. The system is a rather severe test of robustness since only two items have all parame-
ters with values within the ranges used to derive the approximations. There are 10 items with
one extrapolated parameter, 20 items with two extrapolated parameters, 20 with three extrapo-
lations, 10 with four extrapolations, and 2 items with all five parameters extrapolated.
We compare actual values of //, P, R, and Tfor the 64 items with our analytic approxima-
tions. Backlog cost B is not considered because of the complexity of our approximation and the
absence of an explicit dependence on unit penalty cost. The average percent deviations from
actual values of //, P, R, and Tare 1.6%, 0.2%, 1.4%, and 2.6%, respectively. The distributions
of percent deviations are summarized in Table 3. Our approximations are quite accurate con-
sidering the wide range of parameters spanned by the system.
TABLE 3 — Percentage Deviations of Approximations
in a 64-Item System
(Entries are the number of items with errors in the given range,
with the cumulative percentage of items in the system in
parentheses.)
Range of
Holding
Backlog
Replenishment
Total
Deviation
Cost
Protection
Cosi
Cost
[0%,2%)
48 (75%)
^64 (100%)
48 (75%)
30 (47%)
[2%,4%)
6 (84%)
8 (88%)
22 (81%)
[4%,6%)
5 (92%)
0 (88%)
6 (91%)
[6%,8%)
3 (97%)
6 (97%)
4 (97%)
[8%, 10%)
2 (100%)
2 (100%)
1 (98%)
[10%, 12%)
1 (100%)
The holding cost approximation is extremely accurate for all cases with /u greater than 0.5
or aVfi less than 15. All items with deviations greater than 4% have /x equal 0.5 and o-2//jl
equal 15. If we consider only the items with fewer than two parameters extrapolated, the aver-
age error is 0.4%.
The backlog protection approximation is excellent, with only one item having a deviation
in excess of 0.7%.
264
R. EHRHARDT
Our approximation for replenishment cost is also robust. All items with deviations in
excess of 4% have /x equal 0.5, cr2//x equal 15, and K/h equal 16. Items with fewer than two
extrapolated parameters have an average error of 0.1%.
Low fj. and high o-2/(jl are also sources of large errors for our total cost approximation. All
items with deviations in excess of 4% have either /jl equal 0.5 or <r2/fjL equal 15, or both. Items
with fewer than two extrapolated parameters have an average deviation of 1.2%.
We commented in Section 3 that the approximation for total cost may be inaccurate for
items with significantly suboptimal values for 5 and S. The remark is equally valid for the back-
log protection expressions (14) and (15), since they are based on a theoretical result for
optimal policies. This issue is of interest to the analyst who may have reason to use an (s,S)
policy which is designed to satisfy criteria other than simply minimizing total cost. We now
proceed to illustrate how the accuracy of the approximations is affected when nonoptimal values
are used for i and S.
Consider the following system of items that are controlled with nonoptimal policies. We
choose a base-case item with o-2/m equal 5, /jl equal 9, L equal 2, p/ h equal 49, and K/h equal
48. The optimal value of (s,S) for this item is (43,73). We now use this policy on items with
different parameter values. The new parameters are obtained by increasing or decreasing each
base-case parameter value, one at a time, yielding 10 items. The parameter values of the sys-
tem are displayed in Table 4. For each item we compare the actual (exactly computed) and
approximate values of //, P, R, and T.
TABLE 4 — Percentage Errors
of Approximation for Nonoptimal Policies
Percentage Errors of Approximations
Changed
Value
Holding
Cost
Backlog
Protection
Replenishment
Cost
Total
Cost
<r2/fi = 4 (-20%)
6 ( + 20%)
.07%
-.05%
-.6%
.7%
-.00%
.00%
6.0%
-5.0%
fi = 7 (-22%)
11 ( + 22%)
-.11%
-.04%
-1.3%
2.9%
-.03%
.03%
13.7%
-22.2%
L = 1 (-50%)
3 ( + 50%)
-.04%
.02%
-1.6%
5.5%
.00%
.00%
12.6%
-36.2%
plh =39 (-20%)
59 ( + 20%)
-.01%
-.01%
-.5%
.3%
.00%
.00%
3.9%
-2.2%
K/h = 38 (-21%)
58 ( + 21%)
-.01%
-.01%
.0%
.0%
.00%
.00%
4.0%
-2.2%
Average of
Absolute Values
.04%
1.3%
.01%
10.8%
Observe in Table 4 that the approximations for holding cost and replenishment cost are
very accurate, with average percentage deviations of 0.04% and 0.01%, respectively. The
approximation for backlog protection is somewhat less accurate, with the largest errors occur-
ring for large values of lead time and mean demand. The total cost approximation does not
perform well in the system, deviating by an average of 10.8%. Thus, we conclude that the
approximations for backlog protection and total cost should be used with caution for
significantly nonptimal policies. An approach to reducing the errors might be gleaned from the
pattern of deviations in Table 4. Notice that when each parameter is larger than in the base
(s.S) INVENTORY POLICY OPERATING CHARACTERISTICS
265
case, the approximation underestimates the total cost, and when the parameter is smaller than
in the base case, the approximation overestimates the total cost. The reverse is true for backlog
protection.
Finally, we consider the issue of how well the approximations perform when the demand
parameters are not accurately specified. This issue is of interest in applied settings when the
mean /x and variance a-2 of demand are not known but, rather, are estimated using past data.
We have found that the approximations are rather robust when subjected to perturbations of
this type. That is, the relative errors of the operating characteristic approximations tend to be
smaller than the relative errors in the demand parameters. Furthermore, the errors are nearly
symmetric so that when the operating characteristics of several items are aggregated, the errors
due to high values of demand parameters tend to cancel those due to low demand parameters.
As an illustration we consider two items controlled by power approximation policies, one
having a mean demand fx of 4 and the other having /x equal to 12. The other parameters of the
items are identical; demand has a negative binomial distribution with a2//* equal to 5, the lead
time L is 2, the setup cost K is 48, the unit backorder penalty cost p is 49, and the unit holding
cost h is 1. We measure the stability of the operating characteristic approximations by substi-
tuting perturbed demand parameters /jl' and cr' in place of the correct values \x and cr, and com-
paring the approximated values with exactly computed values. For each of the items, we
evaluated the approximations when fi'/fi and cr'2/cr2 took the values 0.80, 0.90, 0.95, 1.00,
1.05, 1.10, and 1.20. All combinations of perturbed values were tested, yielding 49 cases for
each item, or a total of 98 cases.
We summarize the results in Table 5, where average absolute values of relative errors are
listed for several ranges of demand parameter perturbations. Notice that the backlog protection
approximation is not listed in Table 5. This is because the approximation is not a function of
the demand parameters and, therefore, displays no variation when they are changed. The
replenishment cost approximation displays the least stability in Table 5, with an average devia-
tion of 6.9% for the 98 items. Errors ranged up to 19.5% for individual cases with extremely
perturbed demand parameters. The holding cost approximation is more robust, yielding an
average deviation of 4.7% and a maximum deviation of 13.7%. The approximation for total
cost, however, has an average error of only 3.9% and a maximum error of 10.0%.
TABLE 5 — Percentage Errors of Approximation When Demand
Parameters Are Incorrectly Specified
Range for
Demand Parameters
Number
of
Cases
Average Absolute Value of
Percentage Errors
(x'/fJ.
cr'2/*2
Replenishment
Cost
Holding
Cost
Total
Cost
1.0
[.95,1.05]
[.90,1.10]
[.80,1.20]
1.0
[.95,1.05]
[.90,1.10]
[.80,1.20]
2
42
70
98
0.04%
3.0%
4.2%
6.9%
0.2%
2.0%
2.8%
4.7%
1.3%
1.6%
2.3%
3.9%
We note that the data in Table 5 are measures of the accuracy of the approximations for
individual cases. A measure which is perhaps of greater interest in an applied setting is the
aggregate error over all 98 cases, which is less than 0.5% for each of the characteristics. That
is, when the 98 approximated values are averaged and compared with the exact average value,
266 R. EHRHARDT
the difference is less than 0.5%. This observation can be regarded as evidence that the approxi-
mations are relatively unbiased when the demand parameters are replaced with unbiased statis-
tics.
5. CONCLUSIONS
We have derived approximations for replenishment cost (10), holding cost (13), backlog
protection (14), (15), and total cost (17). The expressions are quite accurate and are much
easier to compute than the exact expressions (2). Additional simplification of calculations
could result from using a normal distribution function in lieu of the function G in (12). Then
the six evaluations of G in (12) could be replaced by terms involving the standard normal dis-
tribution function, which requires only a simple table look-up. This possibility has not yet been
investigated.
Despite the good fits obtained in (10), (13), (14), (15), and (17), we caution against their
use in certain circumstances. The results of Section 4 have demonstrated that the approxima-
tions for backlog protection and total cost become less accurate when used for significantly
nonoptimal policies. Although the approximations for replenishment cost and holding cost are
quite accurate over the investigated range of parameter settings, we suspect that they might
break down when used for very small values of D = S - s. This is because (3) is based on an
asymptotic expression for the renewal function M(D).
ACKNOWLEDGMENTS
This paper is based on material from the author's Ph.D. dissertation, written at Yale
University. The author is pleased to acknowledge the guidance and encouragement of his advi-
sor, Professor Harvey M. Wagner. The author also wishes to thank the anonymous referee
whose careful reading of an earlier manuscript helped to improve this paper significantly.
REFERENCES
[1] Ehrhardt, R., "The Power Approximation for Computing (s,S) Inventory Policies,"
Management Science, 25, 777-786 (1979).
[2] Ehrhardt, R., "Operating Characteristic Approximations for the Analysis of (s,S) Inventory
Systems," Technical Report No. 12, School of Business Adminstration and Curriculum
in Operations Research and Systems Analysis, The University of North Carolina at
Chapel Hill, Chapel Hill, N.C. (1977).
[3] Iglehart, D., "Optimality of (s,S) Policies in the Infinite Horizon Dynamic Inventory Prob-
lem," Management Science, 9, 259-267 (1963).
[4] Roberts, D., "Approximations to Optimal Policies in a Dynamic Inventory Model," Studies
in Applied Probability and Management Science, edited by K. Arrow, S. Karlin, and H.
Scarf, (Stanford University Press, Stanford, Calif., 1962).
[5] Smith, W., "Asymptotic Renewal Theorems," Proceedings of the Royal Society (Edin-
burgh), A 64, 9-48 (1954).
[6] Veinott, A. and H. Wagner, "Computing Optimal (s,S) Inventory Policies," Management
Science, //, 525-552 (1965).
OPTIMAL ORDERING POLICIES WHEN ANTICIPATING
PARAMETER CHANGES IN EOQ SYSTEMS
B. Lev and H. J. Weiss
Temple University
Philadelphia, Pennsylvania
A. L. Soyster
Virginia Polytechnic Institute and State University
Blacksburg, Virginia
ABSTRACT
The classical Economic Order Quantity Model requires the parameters of
the model to be constant. Some EOQ models allow a single parameter to
change with time. We consider EOQ systems in which one or more of the cost
or demand parameters will change at some time in the future. The system we
examine has two distinct advantages over previous models. One obvious ad-
vantage is that a change in any of the costs is likely to affect the demand rate
and we allow for this. The second advantage is that often, the times that prices
will rise are fairly well known by announcement or previous experience. We
present the optimal ordering policy for these inventory systems with anticipated
changes and a simple method for computing the optimal policy. For cases
where the changes are in the distant future we present a myopic policy that
yields costs which are near-optimal. In cases where the changes will occur in
the relatively near future the optimal policy is significantly better than the myo-
pic policy.
1. INTRODUCTION
The classical Economic Order Quantity (EOQ) inventory model has several basic assump-
tions that yield the elegant solution of ordering Q* = V2AA7 h where A, K and h are the tradi-
tional inventory parameters of demand, setup and holding, respectively. The most basic
assumption is that all of the parameters are constant. Several systems have been examined in
which either the demand rate or the purchase price may vary with time, (see Goyal [4], Buza-
cott [3], Naddor [9], Resh, Friedman and Barbosa [10], Barbosa and Friedman [1] and Sivazlian
[13].). In all of these papers the parameter changes are continuous with time and furthermore
only one parameter is permitted to change. In this paper we consider EOQ models in which
any or all of the parameters may change at some future point in time.
The system we examine has two distinct advantages over the previous models. One obvi-
ous advantage is that a change in any of the costs is likely to affect the demand rate and we
allow for this. The second advantage is that often, the times that prices will rise are fairly well
known by announcement or by previous experience. If prices have risen January 1, April 1 and
267
268 B. LEV, H. J. WEISS AND A. L. SOYSTER
July 1, it is very reasonable to anticipate a price rise on October 1. Also, price changes are
more likely to jump than to be continuous with time.
In Section 2 of this paper we develop the inventory model and determine the necessary
conditions for a policy to be optimal. In addition, we present a simple method for computing
the optimal policy. Furthermore, a by-product of this method is a myopic policy. The myopic
policy works well when the horizon is large enough and the price or demand change is far
enough in the future. In Section 3, we present computational results for several different sets
of parameters.
2. THE STRUCTURE OF AN OPTIMAL POLICY
Consider a finite horizon of length T that is partitioned into two disjoint time periods; the
closed interval [0,5] called period 1 and the half open interval (S,T\ called period 2. The costs
associated with period 1 are a per unit cost C\, a holding cost rate h\, for all items brought into
stock during period 1 and a setup cost K\ > 0 charged against each order placed during the
period. For items brought into stock during period 2 the unit cost, holding cost rate and setup
cost are c2, h2 and K2, respectively. Thus, 5 is a time at which any or all of the inventory costs
may change. Also, the demand rate may change at 5. Let \i and \2 denote the demand rates
during periods 1 and 2, respectively. A finite sequence of lot sizes is to be purchased to satisfy
the demand. We assume that the initial inventory is zero, delivery is instantaneous, orders are
placed only when the inventory level is zero and the discount factor is either ignored or
included in the holding cost. The optimal policy for cases with a positive initial inventory is
discussed later. Of course, if there are known lead times the results of this paper still hold but
the orders are placed earlier according to the amount of the lead time.
The total cost, Z(Q), for a single order of quantity Q with corresponding holding cost and
purchase cost is Z(Q) = K, + h, Q2/2\, + c,Q. Theorem 1 limits the structure of the optimal
policy as follows:
THEOREM 1: An optimal policy must have the property that
(a) all orders placed and depleted in period 1 are of the same size
and (b) all orders placed and depleted in period 2 are of the same size.
PROOF: Suppose Q\ and Q2 are the sizes of two consecutive orders placed and depleted
in either period and let Q = Q\ + Q2.
The total cost of these two orders Z(QX) as a function of Q\ is given by
Z«2,) = IK, + h,[(Q\)2 + (02)2V2X/ + qiQ, + Q2)
= IK, + h,[Q2 +(Q- Q])2)/2k, + c,Q.
We have that the first and second derivatives are
Z'(Q,)= h,[2Qx- 2(0- <2,)]/2\,
and
Z "(£>,) = 4//,/2A, > 0.
Hence, Z is strictly convex in Q\ and is minimized only at Q\ = ■* = Q2. Thus, two consecu-
tive orders placed and depleted during the same period must be the same size, which implies
OPTIMAL ORDERING IN EOQ SYSTEMS 269
that all orders placed and depleted in either one of these two periods must be the same size,
and the theorem is proved.
Since the orders must be placed and depleted during the same period, Theorem 1 does
not apply to an order that is placed on or before S (period 1) but depleted after 5 (period 2).
Such an order is called a crossing order. Theorem 1 implies that the structure of the optimal
ordering policy has been reduced to one of two possible forms depending on the inventory level
at time S. If the inventory level is zero at 5 (Figure la), then the structure of the optimal pol-
icy is to place m > 0 orders of size Q\ = \\ Sim during [0,5), place an order of size
Qa, 0 < Qa < k2(T — S) at 5, and place n ^ 0 orders of size Q2 = (k2(T - S) - Qa)/n dur-
ing period 2. (Note that if n = 0 then Q2 does not exist). This case is denoted as the zero
inventory case (ZIC). If the inventory level is positive at 5 (Figure lb), then the structure of
the optimal policy is to place m > 0 orders of size Q\ before S, one order of size Qa that
crosses S and n ^ 0 orders of size Q2 after S. This case is denoted as the nonzero inventory
case (NZIC) and the two cases are examined separately.
2.1 Zero Inventory Case
The optimal number of orders to place for the finite horizon inventory model with param-
eters A, //, A", Tis given by Schwarz [12] as the integer n satisfying
(1) nin - 1) < h\T2/2K < n(n + 1).
The right hand inequality is
n2+ n - h\T2/2K > 0.
The solution for the quadratic inequality is
n > - 1/2 + V1/4+ hkP/lK.
The left hand inequality yields
n ^ 1/2 + Vl/4+ hXT2/2K.
Since n is a positive integer
n = < - 1/2 + V1/4+ h\T2llK >
where < x > represents the least integer greater than or equal to x. Define an integer valued
function Nik, /;, K, T) of the inventory parameters as
(2) N(X, //. K, T) = < - 1/2 + Vl/4 4- h\T2/2K >.
(N is used if the parameters are clearly defined).
It follows that for the ZIC the optimal number of orders to be placed during [0,S) is given
by
m* = N(\h //,, Kh S)
and the optimal order size is given by
£i = X,S/m*
The costs incurred in [S, T ] are given by
Fit, n) = Kx + hxk2t2l2 + \2tc\ + n(K2 + h2\2t22l2 + \2t2c2)
270
B. LEV, H. J. WEISS AND A. L. SOYSTER
Inventory
Level
Qa
Q,
\ Q:
Inventory
Level
T time
m orders
■n orders-
a Inventor) al S is zero.
T time
m orders-
n orders
b. Inventory at .V is positive
FlGl ki 1. Optimal Policy Structure-ZIC and NZIC
where / is the length of time it takes to deplete the order placed at S and t2 = (T - S - t)/n.
Letting R = T - S, Fit, n) can be expressed as Fit, n) = A", + hxk2t2/2 + X2/c, +
nK2 + h2\2(R ~ t)2/2n + k2(R - t)c2. The total ZIC costs are thus
(3^ F(t,n) + m*Kx + m*hxQ}/2Kx +klSc] .
The partial derivative of (3) with respect to t, provides a necessary condition for it,n) to
minimize the total inventory costs for the zero inventory case:
(4) 0 = /?,\2/ + \2c, - h2K2{R - t)/n - K2c2
OPTIMAL ORDERING IN EOQ SYSTEMS 271
or t = (n(c2- ci) + h2R)/(nli] + h2).
Notice that if the per unit cost increases then t will be positive. If the cost decreases then t may
be negative. If this is the case then at time S an order should be placed for as few units as pos-
sible or the order should be delayed until time S + e, e > 0.
Also note that, if / is given by (4), then R — Ms the time in which the /; orders are placed and
is given by
R - t= R - (n(c2- cx) + h2R)/(nhx + h2)
= n{hxR - c2 + c,)/(a;/;, + h2).
This will be nonpositive if and only if h\R — c2 + C\ is nonpositive. Thus, if h\R — c2 +
c\ ^ 0, then n must be zero and t = R. This means that if the cost of ordering one unit at
price C] and incurring the holding cost h\ for the entire span R is not more than c2, the incre-
mented purchase price, then obviously one should avoid any purchases at price c2. If
h\R — c2 + C) > 0, then R — Ms positive and n ^ 1. If R — t is positive, then n must be the
optimal number of orders for a finite horizon inventory model of length (R — t). Let
/ = {1,2, ... j and n*(R — t) represent the optimal number of orders to place in the second
period. Then from Equation (1)
n*(R - /) = minU 6 I:n(n + 1) > (X2hj/2K2) (R ~ t)1)
= minU € I.nin + 1) ^ (k^i^lK^ (n(h\R - c2 + cO/inh] + h2))2\
(5) = min{« 6 /:(« + 1) (nh] + h2)2/n > iK2hj2K^ (/;,/? - c2 + c,)2}.
One could compute n* by sequentially searching the integers. However, there exists a more
efficient scheme.
Consider the inequality given inside the braces in (5) expressed as an equality.
Oi/r, + h2)2 Cn + l)//i = (\2h2/2K2) (//,/? - (c2 - c,))2.
Let
z= (X2/?2/2/r2) (/;,/? - (c2- c,))2.
(n2h2 + 2nhxh2 + hj) (« + 1) - nz = 0
n3h2 + 7n2hxh2 + nh\ + n2h\ + 2nhxh2 + /;22 - nz = 0
n3h2 + n2{2hxh2 + h2) + n(h22 + 2/;,/;2 - z) + h\ = 0.
This is a cubic equation and the three roots to the equation can be found using standard alge-
braic techniques (see, for example, Burington [2]). The cubic equation might have a single real
root r\\ or three real roots nu «2, «3, nx < n2 ^ n3. In the former case, the solution to (5) is
n* = <A7i>, and in the latter case, the solution to (5) is
<r\\> if <a?i> ^ n2
</73> if <n\> > n2.
Hence, (5) is easily solvable.
Then
or
or
272 B. LEV, H. J. WEISS AND A. L. SOYSTER
Since the zero inventory case is relatively easy to compute and often performs well as will
be seen in the next section we refer to it as the myopic policy.
2.2 Nonzero Inventory Case
Define /, = Q\/ku t2 = Q2/k2 and ta as the depletion times of the orders placed during
periods 1, 2 and the crossing order respectively. Let m and n be the number of orders placed
during periods 1 and 2 respectively. The total cost is given by
(6) Fin, m, th t2, ta) = m(Kx + /;,\1r,2/2 + c ]X , r, ) + niK2 + h2k2t2/2 + c 2k2t2)
+ K, + /;,{x,(S - w/,)2/2 + iS - mtx)\2{mt\ + ta - S) + (mt] + ta - S^ki/l}
+ (|\,(5 - mt\) + C\k2imt\ + la ~ ^)-
Thus, the mathematical programming problem is:
minimize
Fin, m, t\, t2, ta)
(7)
subject to
mt\ <S
(8)
mt\ + ta >S
(9)
mt\ + ta + ni2 = T
n, m, t\, t2, ta ^ 0
n. m integers.
Notice that due to constraint (9) the problem for a fixed m and /; is a two-dimensional
problem as i2 is determined by the rest of the variables. The problem is still too difficult to
approach as a mathematical programming problem because of the strict inequalities, so we
reduce it to a one-dimensional problem with the following result.
THEOREM 2: For fixed aw, n either <?, = Qa or ZIC is better than NZIC.
PROOF: The proof first shows that when m orders of size £>| are followed by a crossing
order of size Qa then it must be true that Qa = Q\. Let R ^ S be the time at which Qa is
depleted and consider R as fixed. For constants m and R the relationship between Qa and Q\ is
(10) Qa = iS - w^,/\,)\, + (/? - S)\2.
The order, holding and purchasing cost Z for the period [0.R ) as a function of Q\ is
ZiQx) = /*,[mC?2/2A, + iS - mQx/\\)k2iR - S)
+ \,(S - /w0iAi)2/2 + \2iR - S)2/2] + (m + DA', + c,(A,.S + k2iR - S)).
The function is minimized when the first derivative is zero or when
(11) mQi/ki - mk2iR - S)/X, - miS - mQx/k\) = 0.
Notice that the second derivative is im + m2)/k\ > 0 since m > 0. Rearranging (11) yields
(12) (>, = (S- m0,/X,)X, + iR - S)k2.
OPTIMAL ORDERING IN EOQ SYSTEMS 273
This Q\ is the unique optimal order quantity and is equal to Qa from (10) hence all orders are
of the same size.
The decision variable Q\ must satisfy the constraint mQ]/\\ ^ S. If (12) violates this
constraint the solution is on the boundary, i.e., Q\ = h\S/ m which means that all m orders
placed strictly before S are of the same size and the theorem is proved.
We have that
Qa = X, iS - mtx) + X2(mt\ + ta - S)
and from Theorem 2 that Qa = Q\ = \\t\. Hence, it follows that
(13) ta = CM, + (X2 - X,) (S - mtx))l\2.
Furthermore, constraint (8) must be satisfied. Recall that using (13) and (8) one gets
(14) mt] + ta = mtx + (X,f, + (X2 - X,) (S - m/,))/X2
= [im + l)X'ifi + (\2- \X)SV\2-
Notice that mt\ + ta > S if and only if (m + \)t\ > S. Thus, express Fin, m, /i,?2,/0) as a
function of only one depletion time by substituting (13) and t2= (T — imtx + ta))l n into the
expression for Fin, m, t\, t2, ta) given by (6). Denote by f(t\) the cost for a fixed m and n
when the depletion time is t\. Then, after substitution
(15) fUO = im + \)KX + nK2 + ™ ' lh + — 4~ \X2T - im + l)X,r, - (\2 - \,)5]2
2 2nX2
+ ' 1 , — + hxiS - mr,)Xi [im + 1)^ - S) + -^-[(m + Or, - S]2
2 2A. 2
+ c,\ir,(m + 1) + c2k2T - (m + l)c2Xi^i - c2(X2 - X,)5.
Now /'(/i) is given by
(16) /'(/,) - A^^ 7-W2T- im + 1) \,r, - U2 - XiJ^U^m + 1)
nk2
h X2
- /?,X, [(/w + l)r, - S]m + -J-1 [(m + Of, - S](m - 1) + X,(m + 1) ic{ - c2).
X2
Also,
(17) f"itx) = im + 1)2X2 [hj/n - /7,]/X2 - h^mim + 1).
Now if (17) is negative /(■) is concave and hence the minimum occurs at an extreme point of
the feasible region. Thus, either the minimum is a zero inventory case or t = T/in + m + 1).
If (17) is positive /(•) is convex and either the minimum is at an extreme point and again we
have the zero inventory case or t = T/in + m + 1) or the minimum occurs by setting the
derivative equal to zero. This leads to the following:
THEOREM 3: If for a fixed m and n the optimal case is the nonzero inventory case then
either t] = T/im + n + 1) or
h2\2T- ih2 + «/?,)(X2- X,)S + nk2ic2- c,)
(18) u =
im + \)\\ih2 + nh\) — nmh\\2
274 B. LEV, H. J. WEISS AND A. L. SOYSTER
Notice that if there are no changes then t\= T/k where A: is the number of orders that
are of the same size as previously shown by Schwarz [12]. Also, if only the demand changes
then ta = t2. Given /,, the last task is to find m and //. As before, if t\ is the depletion time of
each of the first m orders, then T — mt\ — ta is the length of time for the last n orders and the
optimal number of orders placed during [mt\ + ta, T] must satisfy Equation (1). That is,
n*(T- mt] - ta) = min [n G l.nin + 1) ^ iXihJlk^ (T - mtx - ta)2).
It appears that one needs to compute t\ and // for all values of m. This would be a for-
midable task. However, the number of possible values for m can be reduced by the following:
THEOREM 4: For the case where the inventory level is positive at S either m* = NiS)
or m* NiS) + 1 where NiS) is the optimal number of orders to place in a finite horizon
m + 1
[0, S]. Furthermore, n* ^ N
S
m
PROOF: Let a = mtu S < a < T. a > S — X(a) > N(S), since (2) is nondecreas-
ing. All orders must be placed before S. Let b be the time of the last order. Then
Nib) < N(S), hence, m* < Nib) + 1. Thus, either NiS) or NiS) + 1 orders are placed.
The restriction on n* follows from Theorem 3 in [5].
We now can solve the NZIC for w = NiS) and for m = NiS) + 1 and take the
minimum cost of the ZIC and the NZIC. The algorithm is as follows:
1. Calculate NiS) from (2) and set m = NiS).
2. Calculate NiT- S) from (2) and set n = NiT - S).
3. Calculate t*in) from (4) and compute the cost for the ZIC from (3).
T_»L+AS
m
C from (15).
to NiT ' — S) calculate ft im, n) from (18) and the cost
4. For n = N
for the NZ
5. Set m = NiS) + 1.
6. Repeat step 4.
7. Find the minimum costs from steps 3,4,6
The last detail to discuss is that of an initial inventory. If the beginning inventory, /0, is
less that or equal to k\S, obviously the inventory should be depleted and the problem is that of
a finite horizon of length T - /q/A, with a price change at time 5 - /(Ai- If the beginning
inventory will not be depleted until after time 5, obviously no purchases should be made until
at least time 5". In this case, the cost of not purchasing at S and then purchasing when the
inventory is depleted should be compared with the cost of purchasing units at time S.
3. COMPUTATIONAL RESULTS
It is interesting to determine what effect varying the horizon or the time at which the
parameters change would have on the optimal policy. In particular whether or not the myopic
OPTIMAL ORDERING IN EOQ SYSTEMS
275
zero inventory case is optimal and if not how close to optimal it is. Note that for the case of no
changes the optimal cost as a function of the horizon appears as in Figure 2 (see [5], [12]).
Schwarz [11] has shown that if the horizon is at least 5 EOQs worth then the optimal finite hor-
izon cost is no more than 1% above the optimal infinite horizon cost. One expects similar
behavior in this model.
s/ZXEh
time
Figure 2. Optimal cost as a function of time when parameters remain constant
Table 1 contains the optimal costs for both the zero inventory case and nonzero inventory
case where all parameters are fixed except for the horizon. The per unit cost was changed by .1
and the holding cost by .025. The demand and setup cost are constant throughout the two
periods. Notice from Table 1 that the optimal policy alternates back and forth between the
myopic and nonmyopic policies. Also, as the horizon becomes large the overcost when using
the myopic policy tends to decrease. In fact, for any horizon above 25 the overcost is less than
1%. Incidentally, the infinite horizon optimal policy is the zero inventory case, with an average
cost of 50.75.
TABLE 1 — Inventory Costs as a Function of Horizon Length
for AC = .1 (2%) and Ah = .025 (2%)
k= i
, A', = K2 = 50, /?, =
1.25, h2= 1.275, c, =
5, c2= 5.1, S= 20
T
ZIC
Average Cost
NZIC
Average Cost
ZIC/NZIC-1
21
53.66
50.02
7.28%
22
52.58
50.09
4.97
23
51.74
50.02
3.44
24
51.09
50.00
2.18
25
50.63
50.14
.98
26
50.32
50.17
.30
27
50.15
50.12
.06
28
50.11
50.40
—
29
50.17
50.30
—
30
50.34
50.23
.22
31
50.60
50.19
.82
32
50.19
50.41
—
33
50.23
50.33
—
34
50.34
50.29
.10
35
50.26
50.25
.02
36
50.25
50.42
—
37
50.28
50.36
—
38
50.36
50.32
.08
39
50.30
50.30
—
40
50.30
50.44
—
276
B. LEV, H. J. WEISS AND A. L. SOYSTER
Table 2 contains similar information but for a larger price increase. Let AC= 1 and
A/j = .25 while all other parameters are as above. Again, when the horizon is 25 or larger the
myopic policy is never worse than 1% above optimal. However, in this case the myopic policy
is optimal for all horizons larger than 35.
TABLE 2 — Inventory Costs as a Function of Horizon Length
for^C= 1 (20%) andkh= .25 (20%)
A =
5, Kx= K2= 50, ft,=
= 1.25, h2= 1-5, C,=
5, C2= 6, S =
20
T
ZIC
Average Cost
NZIC
Average Cost
ZIC/NZIC-1
21
53.71
50.02
7.38%
22
52.75
50.09
5.31
23
52.01
50.02
3.98
24
51.48
50.00
2.96
25
51.12
50.63
.97
26
50.93
50.90
.06
27
50.87
51.23
—
28
50.94
51.38
—
29
51.12
51.61
—
30
51.41
51.90
—
31
51.79
51.99
—
32
52.24
52.11
.25
33
51.89
52.44
—
34
52.12
52.50
—
35
52.41
52.58
—
36
52.34
52.89
—
37
52.50
52.93
—
38
52.69
52.99
—
39
52.74
53.06
—
40
52.85
53.30
—
In the examples presented in Table 3 the horizon is fixed and the time of price change
varies. The remaining parameters are identical to those of Table 1. The Table also contains
which case is optimal in the long run. Notice how in the infinite horizon model as in the finite
horizon model the cases alternate as S changes. Also, as S approaches Fthe myopic policy wor-
sens.
In the next example presented in Table 4, S varies, and we use the larger cost increase as
in Table 2. This time, the infinite horizon models always are optimized by the myopic policy.
Again, as S approaches Tthe myopic policy begins to worsen.
The last set of examples given in Table 5 indicates that as the number of orders (using
either policy) increases then the difference between the myopic and optimal policies lessens.
The data used to generate Table 5 is identical to the data for Table 1 except that the holding
cost is reduced from 25% of the purchase cost to 5% of the purchase cost. Notice that this gen-
erates fewer orders which in turn increases the overcost.
OPTIMAL ORDERING IN EOQ SYSTEMS
277
TABLE 3 — Inventory Costs as a Function of the Time of Price
Changes for AC= .1 (2%) and kh = .025 (2%)
\ = 5, Kx= K
2= 50, /?,= 1.25, h2= 1.275, c, = 5,
(and T = °° for last column)
c2= 5.1, T= 30
S
ZIC
NZIC
ZIC/NZIC-1
Optimal Case
6
50.71
50.60 1 2 5*
50.60 ) '
.22%
NZIC
7
50.54
—
ZIC
8
50.54
50.50 3,5
.06
ZIC
9
50.50
50.50 |
—
NZIC
10
50.54
50.50 | 3,4
.08
NZIC
11
50.43
50.50 J
—
ZIC
12
50.45
50.41
50.41 J
.04
ZIC
13
50.38
—
NZIC
14
50.40
50.38 1 43
50.38 J '
.04
NZIC
15
50.33
—
ZIC
16
50.38
50.32 |
50.32 (
.12
ZIC
17
50.28
—
NZIC
18
50.28
50.27 , 9
50.27 J '
.02
NZIC
19
50.64
.74
ZIC
20
50.34
50.23 , ~
50.23 J 6'2
.22
ZIC
21
50.19
—
NZIC
22
50.17
50.16 6,1
.02
NZIC
23
50.15
50.14 )
.02
ZIC
24
50.28
50.14 [ 7,1
.28
ZIC
25
50.54
50.14 J
.80
NZIC
The notations s
hould be read as follow:
The optimal p
Dlicy for S = 6 and S=l is m=2n =
5
278
B. LEV, H. J. WEISS AND A. L. SOYSTER
TABLE 4 — Inventory Costs as a Function of the Price Changes
for AC = 1 (20%) and A/? = .25 (20%)
\ =
5, K\ = K2= 50.
h] = 1.25, h2= 1.5, c, = 5, c2 = 6. T =
30 (and T = oo )
5
ZIC
NZIC
ZIC/NZIC-1
T = oo
6
55.00
55.16
—
ZIC
7
54.59
55.16
—
ZIC
8
54.50
55.16
—
ZIC
9
54.13
54.71
—
ZIC
10
53.93
54.20
—
ZIC
11
53.60
54.00
—
ZIC
12
53.40
54.22
—
ZIC
13
53.11
54.22
—
ZIC
14
52.91
53.26
—
ZIC
15
52.84
52.91
—
ZIC
16
52.41
52.91
—
ZIC
17
52.11
52.93
—
ZIC
18
52.48
52.38
—
ZIC
19
51.87
51.89
—
ZIC
20
51.41
51.56
—
ZIC
21
51.11
51.56
—
ZIC
22
50.95
50.94
—
ZIC
23
50.80
50.94
—
ZIC
24
50.80
50.78
.04%
ZIC
25
50.95
50.63
.64
ZIC
TABLE 5 — Inventory Costs as a Function of Horizon Length
for±C= .1 (2%) and Mi = .005 (2%)
X = 5,
A", = K2 = 50. /?,=
.25. /;2= .255. c,=
5. r2= 5.1, S= 20
T
ZIC
Average Cost
NZIC
Average Cost
ZIC/NZIC-1
21
40.50
36.32
11.50%
22
39.85
36.40
9.47
23
39.28
36.31
8.18
24
38.79
36.25
7.01
25
38.36
36.20
5.95
26
37.99
36.18
4.99
27
37.67
36.18
4.12
28
37.40
36.19
3.34
29
37.16
36.21
2.62
30
36.97
36.25
1.98
31
36.80
36.50
.99
32
36.66
36.39
.74
33
36.56
36.36
.54
34
36.47
36.34
.37
35
36.42
36.33
.24
36
36.38
36.32
.14
37
36.36
36.33
.07
38
36.36
36.35
.02
39
36.37
36.37
<01
40
36.40
36.48
—
OPTIMAL ORDERING IN EOQ SYSTEMS 279
In summary, if the horizon is large, compared with the time of price change (we suspect
that large is 5 EOQs) then the myopic policy appears to be very worthwhile.
ACKNOWLEDGMENT
We thank the anonymous referee for his suggestions and corrections.
BIBLIOGRAPHY
[1] Barbosa, L.C. and M. Friedman, "Deterministic Inventory Lot Size Models— A General
Root Law," Management Science 24, 819-826 (1978).
[2] Burington, R.S., Handbook of Mathematical Tables and Formulas, 4th Edition (McGraw-
Hill, New York, N.Y. 1965).
[3] Buzacott, J. A., "Economic Order Quantity with Inflation," Operational Research Quarterly,
26, 3 (1975).
[4] Goyal, S.K., "An Inventory Model for a Product for which Purchase Price Fluctuates,"
New Zealand Operational Research, 3, 2 (1975).
[5] Lev, B. and A.L. Soyster, "Inventory Models with Finite Horizons and Price Changes,"
Operational Research Quarterly, 30, 1, 43-53 (1979).
[6] Lev, B., H. J. Weiss and A.L. Soyster, "Comment on an Improved Procedure for the Fin-
ite Horizon and Price Changes Inventory Model," Operational Research Quarterly, 30,
9, 840-842 (1979).
[7] Lippman, S.A., "Economic Order Quantities and Multiple Set Up Costs," Management Sci-
ence, 18, 39-47 (1971).
[8] Ludin, R.A. and T.E. Morton, "Planning Horizons for the Dynamic Lot Size Model: Zable
vs. Protective Procedures and Computational Results," Operations Research, 23, 711-
734 (1975).
[9] Naddor, E., Inventory Systems, 48-50 (John Wiley and Sons, New York, N.Y. 1966).
[10] Resh, M., M. Friedman and L.C. Barbosa, "On a General Solution of the Deterministic
Lot Size Problem with Time Proportional Demand," Operations Research, 24, 718-725
(1976).
[11] Schwarz, L.B., "A Note on the Near Optimality of ^-EOQ's Worth' Forecast Horizons,"
Operations Research, 25, 533-536 (1977).
[12] Schwarz, L.B., "Economic Order Quantities for Products with Finite Demand Horizons,"
AIIE Transactions 4, 234-237 (1972).
[13] Sivazlian, B.D. and L.E. Stanfel, Analysis of Systems in Operations Research (Chapter 5),
(Prentice Hall, Englewood Cliffs, N.J. 1975).
SYSTEMS DEFENSE GAMES:
COLONEL BLOTTO, COMMAND AND CONTROL*
Martin Shubik
Yale University
New Haven, Connecticut
Robert James Weber
Northwestern University
Evanston, Illinois
ABSTRACT
The classical "Colonel Blotto" games of force allocation are generalized lo
include situalions in which there are complementarities among the targets being
defended. The complementarities are represented by means of a system
"characteristic function," and a valuation technique from the theory of coopera-
tive games is seen to indicate the optimal allocations of defense and attack
forces. Cost trade-offs between systems defense and alternative measures,
such as the hardening of targets, are discussed, and a simple example is
analyzed in order to indicate the potential of this approach.
1. COLONEL BLOTTO GAMES
The first example of what has come to be called a "Colonel Blotto game" was apparently
given by Borel [3]. He discussed the case of a defender attempting to protect several locations
against an aggressor. A typical objective of the aggressor was to maximize the expected number
of locations captured.
Games involving this type of objective were subsequently studied by Tukey [11] and oth-
ers (for example, Gross [7], Blackett [2], Dresher [4], Beale and Heselden [1]). As defined by
Beale and Heselden, a (Colonel) Blotto game is a zero-sum game involving two opposing
players, I and II, and n independent battlefields. I has A units of force to distribute among the
battlefields, and II has B units. Each player must distribute his forces without knowing his
opponent's distribution. If I sends xk units and II sends yk units to the Arth battlefield, there is
a payoff Pk(xk,yk) to I as a result of the ensuing battle; the payoff for the game as a whole is
the sum of the payoffs at the individual battlefields.
In this paper we consider a generalization of the classical Blotto game. This generalization
gives regard to the important class of military problems wherein there exist complementaries
among the points being defended. In such cases, the final status of the competitors is not
determined merely by totalling individual target values, but depends on the relative value of
'This research was supported in part by a contract with the U.S. Office of Naval Research.
281
282 M. SHUBIK AND R. J. WEBER
capturing (or neutralizing) various configurations of targets. Our generalization includes the
classical Blotto games, as well as, for example, games in which the aggressor's objective is to
maximize the probability of capturing a majority of the targets.
By considering complementarities among targets, we are in a position to study the defense
of networks. For the purposes of increased reliability and security, redundancy is often inten-
tionally incorporated into telephone and electrical power grids, early warning networks, and
command and control systems. It is natural to ask how well protected such systems are from a
disabling attack. Furthermore, it is of interest to consider cost trade-offs between built-in
redundancy and extrinsic defense. In order to pursue these issues, we first introduce some ter-
minology from cooperative game theory.
2. SYSTEMS PERFORMANCE AND THE CHARACTERISTIC FUNCTION
An w-person game in coalitional form is described by a characteristic function v() defined
for all subsets of the set N of "players." When one is considering networks (or battlefields, or
strategically important facilities), v(S) may be interpreted as the value remaining in the system
if only the set of nodes S is held. The characteristic function captures in a general setting the
many types of complementarity which can exist among the various combinations of points in
the network. (In traditional cooperative game theory it is frequently assumed that the charac-
teristic function is superadditive; that is, if S and T are disjoin*, then v(S) + v(T) <
v(5 U T). However, in the context of strategic systems this assumption may not be reason-
able. If one is protecting a network of doomsday devices, for example, the characteristic func-
tion might assign a value of 1 to every nonempty set.)
There are many different "solutions" which have been suggested by game theorists for
games in coalitional form. They reflect various aspects of the cooperative dealings among
players with different goals. We note in particular the value solutions, which can be given an
interpretation in terms of the military problem of allocating forces to a system of n nodes. In
order to give this interpretation in detail we must reformulate the original //-person game as a
two-person noncooperative game.
3. THE NONCOOPERATIVE GAME
We recast the given game as if it were a zero-sum game played between two opponents, a
defender and an attacker. The n players in the original game are regarded as nodes (or indivi-
dual targets) in a strategic network that the defender is trying to protect and the attacker is try-
ing to destroy.
Let A and B be the respective amounts of strategic resources (troops, for example, or
antiballistic and ballistic missiles) held by the defender and the attacker. The defender may
choose any nonnegative allocation x = (x{, . . . , x„) of resources, subject to the constraint that
Ix,= A. Similarly, the attacker may choose any allocation y = (y, yn) for which
I yt = B. Let J)(xl,yi) be the function (yet to be specified) which indicates the outcome of the
battle at point ./. A natural interpretation which we take at this time is that fji.Xj.yj) is the pro-
bability that the defender retains point ./.
Assume that the goal of the defender is to maximize the (expected) effectiveness of the
surviving configuration of targets. If the interests of the attacker are directly opposed to those
of the defender, then we have at hand a two-person zero-sum game. The probability that the
targets in the set S survive, while all others are destroyed, is
COLONEL BLOTTO, COMMAND AND CONTROL 283
Therefore, the expected effectiveness of the surviving collection is
Sc/V
this is the defender's payoff.
If we suspend the interpretation of the functions f, as probabilities, we find that this com-
petitive game is indeed a direct generalization of the traditional Colonel Blotto game. Assume
that the underlying characteristic function is additive, so that v(S) = £ v({*}) for all S N.
Then
D(x,y)= £ fk(xk,yk) v({k})
A-6S
k=\
By identifying Pk(xk,yk) with fk(xk,yk) • \{{k}) ( for example, by taking Pk = fk and
v({A:)) = 1 for all k € A7), we can represent any desired classical Blotto game.
4. BATTLE MODELS
A listing of the various battle models which have been considered is beyond the scope of
this paper. Moreover, a critical evaluation of the relative validity of these models does not
appear to be available. Even Napoleon's dictum that God is on the side of the strongest bat-
talion does not appear to be borne out when the force sizes of victors and losers in major bat-
tles are compared (for example, see Dupuy, page 89 [6]).
For the purposes of this paper we have chosen to consider a moderately general class of
models in which the attacker and defender have homogenous resources. Hence, force mix
problems have been set aside. Still, while it may be reasonable to assume that the probability
that a target j is captured or destroyed is simply a function fl(xj,yl) of the resources expended
in attack and defense by the two sides, the actual form of this function depends on empirical
factors such as target type, physical vulnerability, troop morale, and the like.
We specifically consider outcome functions of the form
f(.x,y) = ,
yxm + (\-y)ym
where we set /(0,0) = y. The parameter y may be interpreted as an indicator of the natural
defensibility of the target; if x = v, then f(x,y) = y. The homogeneity of the function /allows
us to concern ourselves with the ratio k = x/y of defending to attacking forces, rather than
with the specific amounts x and y. The parameter m reflects the importance of the relative
difference in size between the attacking and defending forces.
In the limit, as m becomes large, the outcome function becomes the crudest form of
"superior forces" model: the side which commits a greater force will win with certainty. If the
resources of the defender and the attacker are of comparable size, in this limiting case the
force-allocation game may fail to have a solution in pure strategies. (For an investigation of the
degree of disparity of initial force sizes sufficient to guarantee the existence of optimal pure
strategies, see Young, [13]).
284 M. SHUBIK AND R. J. WEBER
On the other hand, if m is not too large, the outcome function is relatively insensitive to
small changes in opposing allocations. We consider this case in the next section.
5. VALUE SOLUTIONS
Let v() be a characteristic function on N, and let p = (p\ p„) be a vector of proba-
bilities (that is, each 0 ^ p, ^ 1). Then the (p\ p„)-value of v is the /?-vector
j8 = (0| /8 „) defined for all / € N by
' (v(S U /) - v(S)].
0f- I
SC V,,
ru n n-p*)
/ € S k 6 A \S
Consider the force-allocation game based on v, in which the initial resources of the oppos-
ing sides are A and B, respectively. Assume that the outcome function at the /cth target is
defined by fk(x,y) = y kx'"/(y kx'" + (1 — yk)ym). Then if both sides have optimal pure stra-
tegies, these strategies must be force allocations proportional to the (/] ./„) (,4,fi)-value
of the underlying game. Furthermore, for all sufficiently small values of w, allocations propor-
tional to the (f\, ... , /„) (A,B)-va\ue are indeed optimal.
Further details concerning these results are presented elsewhere (Shubik and Weber [9]).
6. THE COSTS OF SYSTEMS DEFENSE
"What price freedom?" is an important question, but one which political philosophers,
economists, and Department of Defense budget proposers often find difficult to make precise.
A model which links the value and cost of defense is presented here. (A different model is
presented in Section 7, where we take the cost of defense as given but consider the possibility
of trade-offs between direct defense and the physical reinforcement of individual targets.)
At an abstract level, there are four major items in the description of a defensive system:
the military or societal "worth" of defense; the type, quantity, and structure of defensive forces;
the cost of these forces; and the "hardness" (defensive strength) of individual targets.
The model of Section 3 avoids the problem of comparing value and cost by representing
value within the characteristic function and taking as given the available attack and defense
forces. Thus, constraints on military resources enter only as boundary conditions on a force
assignment problem, rather than as a result of taking resource costs into account in the payoff
structure.
We can modify the games of Section 3 to include costs in the following manner. The
defender and attacker first select force levels k\ and fc2, incurring costs of C\(k\) and c2(k2).
They then each assign forces, and the payoffs are given by
(*) Pn= v(S) - c,(Ar,), and
Px = w(S') - c2(k2),
where v(S) is the worth (in monetary units) to the defender of the configuration S of surviving
targets, and w(S') is the worth to the attacker of destroying or capturing the targets in S' . This
is a two-stage nonconstant-sum game, which might be studied in terms of either equilibrium or
minimax theories.
COLONEL BLOTTO, COMMAND AND CONTROL 285
The fact that the above game formulates well as a two-stage process calls attention to the
fact that the two stages are separate in both time and bureaucratic control. The problem for a
defense department in dealing with the government as a whole is to select kh incurring the
budgetary expense c,(/c,). The problem of the commander, having been presented with forces
A',, is to allocate these forces wisely.
From the viewpoint of analysis, the models of Section 3 seem worth pursuing at the level
of command and control. However, it appears that the first stage of the model suggested by (*)
concerns a very different aspect of decision making, and involves deep issues in the area of
defense budgeting (some of these issues have been discussed by Hitch and McKean [8]).
7. THE HARDENING OF TARGETS
In order to illustrate some of the preceding considerations, we analyze a simple example.
Assume that a defender seeks to protect three sites, at each of which several antiballistic mis-
siles are siloed. If the attacker destroys any two (or all three) of the targets, the overall defen-
sive system will collapse. The first site houses more missiles than the second, which in turn
houses more than the third; although any two surviving sites will yield an adequate system, the
survival of all three provides even greater security. We model this situation with a characteris-
tic function v, which satisfies v(123) = 4; v(12) = 3; v(13) = 2; v(23) = 1; and v(S) = 0 if
\S\ < 1.
Assume that the attacker and defender possess comparable amounts of strategic resources;
say, A = B = 1. Let the outcome of conflict at site k be represented by the function fk(x,y) =
y kxl("Y k*™ + (1 — yk)ym), f°r some moderately small value of m (that is, assume that equal
forces engaged at site k will yield a result favorable to the defender with probability yk, and
further assume that small differences in resource assignments lead to only relatively small
changes in this probability). The parameter yk indicates the "hardness" of the target at site k
(that is, its natural strength against attack). It follows, as was indicated in Section 5, that the
optimal allocation of strategic forces by each side will be proportional to the (y |(y2,y3)-value of
the game v. Hence, this allocation will be proportional to the vector
P = (2y3 + 3y2 - 2y2y3, 3y , + y3 - 2y,y3> 2y, + y2 - 2y,y2).
In particular, if we initially have yi = y2= y.i = 1/2, the optimal allocation for each side is
(4/9, 3/9, 2/9).
Now, assume that additional capital is available to the defender, and may be used to har-
den any of the targets. Specifically, assume that an investment of AcA units of capital at site k
will yield an increase of (1 — yk)kck in the hardness of target /r, that is, dyk/dck = (1 — yA).
A natural question is how best to invest the additional capital.
Let the defender's allocation of forces be x = (x^x^Xj), while the attacker's deployment
is y = (y\,y2,.V})- Then the value of the outcome of the competitive game, to the defender, is
D(x,y) = 3/,/2 + 2/,/3 + f2f} - 2/,/2/3,
where each fk is evaluated at (xk,yk). The optimal strategies are x* = y* = j8/L /3,. Therefore,
the rate of gain from investment in the hardening of target k is
(x*,y*) = - — (x*,y*) - — (x*,y*)
dck dpk dyk dck
= 08*710,) ■ 1 • (1 -yk).
286
M SHUBIK AND R J. WEBER
The best investment is in the target (or targets) for which this expression is maximized. But
the expression varies with the parameters y t , y2-. ar>d 7v Hence, if we begin with all yk equal,
it is best to initially invest in work at the site for which fik is maximal; this changes /3 as well as
yk, after which we can determine the best target for further investment. Beginning with
yi = 72 = 73 = 1/2, we obtain the results indicated in the figures. (As the available capital
increases without limit, the value of D(x*,y*) approaches 4, and the three sites attract nearly
equal proportions of the capital.)
This example illustrates several, but by no means all, of the types of computations which
appear to be feasible and relevant to the study of tradeoffs in defense, in the hardening of tar-
gets, and in built-in system redundancy.
100%
67% - -
33% - -
5 10
I igi ki I Allocation ol capital to large! reinforcement
\ — CAPITAL
CAPITAL
FlGURl 2 Hardness of targets y: y: ,uul y.
COLONEL BLOTTO, COMMAND AND CONTROL
287
\ — CAPITAL
5 10
Figure 3. Value of game lo defender: D(\*. v*).
BIBLIOGRAPHY
[1] Beale, E.M.L. and G.P.M. Heselden, "An Approximate Method of Solving Blotto Games,"
Naval Research Logistics Quarterly 9, 65-79 (1962).
[2] Blackett, D.W., "Pure Strategy Solutions to Blotto Games," Naval Research Logistics Quar-
terly 5, 107-109 (1958).
[3] Borel, E., Traite du calcul des probabilites et des ses applications, Applications des jeux de
hasard. Vol. IV, Fascicule 2, (Gauthier-Villars, Paris, France, 1938).
[4] Dresher, M., Games of Strategy: Theory and Applications, (Prentice Hall, Englewood Cliffs,
N.J., 1961).
[5] Dubey, P., A. Neyman and R.J. Weber, "Value Theory without Efficiency," Mathematics
of Operations Research (to appear).
[6] Dupuy, T.N., "Analyzing Trends in Ground Combat," History, Numbers, and War, /, 2
79-91 (1977).
[7] Gross, O. and R. Wagner, "A Continuous Colonel Blotto Game," RAND Memorandum
408 (1950).
[8] Hitch, C.J. and R.M. McKean, The Economics of Defense in the Nuclear Age (Harvard
University Press, Cambridge, Mass., 1960).
[9] Shubik, M. and R.J. Weber, "Competitive Valuation of Cooperative Games," Mathematics
of Operations Research (to appear).
.10] Shubik, M. and H.P. Young, "The Nucleolus as a Noncooperative Game Solution," Cowles
Foundation Discussion Paper No. 478, Yale University (1978).
[11] Tukey, J.W.,"A Problem of Strategy," Econometrica 17, 73 (1949).
L12] Weber, R.J., "Probabilistic Values for Games," Cowles Foundation Discussion Paper No.
471R, Yale University (1978).
[13] Young, H.P., "Power, Prices and Incomes in Voting Systems," International Institute for
Applied Systems Analysis RR-77-5, March 1977.
ON NONPREEMPTIVE STRATEGIES IN
STOCHASTIC SCHEDULING
K. D. Glazebrook
University of Newcastle upon Tyne
Newcastle upon Tyne, England
ABSTRACT
It is shown that there is an optimal strategy for a class of stochastic schedul-
ing problems which is nonpreemptive. The results which yield this conclusion
are generalizations of previous ones due to Glazebrook and Gittins. These new
results also lead to an evaluation of the performance of nonpreemptive stra-
tegies in a large class of problems of practical interest.
1. INTRODUCTION
A job shop consists of one machine and a set J = {1, 2 K] of jobs to be processed
on it. In general the processing time P, for job / is a positive integer-valued random variable
with known honest distribution, processing times for different jobs being independent. If job /
is completed at time F, (flow time) its cost is C,(F,). There is a precedence relation R on the
set J such that if (/', j) € R then the machine must complete job / before it can begin process-
ing job j.
For simplicity, the major part of the material will be devoted to problems in discrete time.
During each time interval [t, t + 1), / € Z+ U {0}, just one of the unfinished jobs is processed
by the machine. A feasible strategy tt is any rule for deciding how to choose the jobs in J for
processing which is consistent with R. Under strategy tt job / is completed at the random time
Fjbr). The objective is to find those strategies tt in some given subset of the set of feasible
strategies which minimize the total expected cost
TC(tt) = E
£c,[F»]
The economic criteria which have been most widely studied in this context are the
discounted costs criterion, that is
(1) C,(F,)- - K(i)aF', {0 < K(i)}, (0 < a < 1), i € J
(see [1], [2], [3], [4], [6], and [9]), and the criterion involving linear costs, that is
(2) C,{F,)= K(i)F„ {0 < AT (/).}, / € J
(see [2], [3], [6], and [10]).
289
290 K. D. GLAZEBROOK
The problem of finding optimal feasible permutations of J for the above economic criteria
has essentially been solved in the sense that algorithms have been given which can be shown to
generate all the optimal permutations. For details of this work see [4], [6] and [10]. Much
work, however, remains to be done on the efficiency of these algorithms.
The problem of finding strategies which are optimal in the set of all feasible strategies for
economic criteria (1) and (2) is much more difficult. Glazebrook [2] gave a characterization of
the optimal strategies for the case when R has a digraph representation which is an out-tree.
Results in a similar vein, though obtained in a rather different way, were reported by Meilijson
and Weiss [5]. The problem with general R seems very complex.
Not surprisingly, then, concentration has latterly focused on the problem of giving a char-
acterization of those problems which have an optimal strategy (in the set of all feasible stra-
tegies) given by a fixed permutation of / For contributions in the vein, see Glazebrook [3]
and Glazebrook and Gittins [4]. All the results known in this area to date require that in some
sense the future prospects of the jobs improve indefinitely as they are processed. For example,
Glazebrook and Gittins prove that when the function
(3) E(aP' X\P, > x + 1)
is nondecreasing in x for each i € J (this happens if P, has a nondecreasing hazard rate) there
is an optimal strategy for economic criterion (1) given by a fixed permutation of / However,
in many contexts, for example research planning (see Nash [7]), it is rather more realistic to
expect that the future prospects of jobs, after an initial (perhaps lengthy) period of improve-
ment, will begin to deteriorate. It is with this in mind that in Section 2 we demonstrate that
the above result of Glazebrook and Gittins may be generalized in a way which does not put
monotonicity requirements on the function in (3). Some extensions of this result are discussed
in Section 3. In Section 4 we demonstrate how the results of Section 2 may be utilized to give
an indication of how well an optimal permutation performs relative to an optimal strategy in a
wide range of problems of practical interest. We conclude in Section 5 with a simple example
involving five jobs.
2. THE MAIN RESULT
We shall consider the problem of finding optimal strategies (in the set of all feasible stra-
tegies) for the pair (J, R) when the economic criterion (1) applies. We shall demonstrate that
there exists an optimal strategy which is deterministic, stationary, Markov and nonpreemptive
(DSMNP), that is which is given by a fixed permutation specifying in which order the jobs are
to be done, when the following conditions hold:
CONDITION 1: m(i, x) > m(i, 0), x € Z+, i € J-
CONDITION 2: lim m(i, x) exists and is strictly greater than mU, 0), i € J\ the func-
X— "«o
tion m(i, .) being defined as follows:
p(P, > x + 1) > 0-£> m(i, x) = E(aFr*\P, > x + 1);
p(P, ^ x + 1) = 0=> mU, x) = 1.
STRATEGIES IN STOCHASTIC SCHEDULING 29 1
Conditions 1 and 2 are more general than those given by Glazebrook and Gittins [4].
Condition 1 states (loosely) that a task is always brought nearer completion by being processed
for an arbitrary length of time. As will be demonstrated in Section 4, the results of this section
have implications beyond problems in which conditions 1 and 2 are satisfied.
Before proceeding to the proof of our main result, note first that it is a consequence of an
important result in the theory of Markov Decision Processes (see, for example, Ross [8]) that
there exists an optimal strategy for our problem which is deterministic, stationary and Markov
and so we may restrict our analysis to such strategies.
We require some terminology and notation. By the state of an incomplete job we mean
the amount of processing it has received. If job / has been completed its state is denoted *,.
We denote by C(x\, x2 xk) = C(x), the total expected cost incurred by all the jobs in a
system identical to the one under study except that job j is in general state x, initially instead of
necessarily being in state 0, j 6 J, the assumption being that an optimal strategy is adopted.
C(x) is similarly defined, the assumption now being that an optimal DSMNP strategy is
adopted. / denotes the subset of jobs in J which have no predecessors according to R, i.e.,
/ = {/; / € J and 0', /) £ R for any j € J).
Both C(x) and C(x) may be characterized as the solutions to appropriately formulated
dynamic programming optimality equations:
C(x) = min [ap(P,= x, + l|P, > x,){— K(i) + C(x,, ... , x,_,, *, x/+1, .... xK)}
;€/
and
+ ap(Pj > Xj + \\P, > Xj)C(x\, ... , x,_1( Xj + 1, xj+\, ... , xK)],
C(x) = min [— K(i)m(i, x,) + m (/, x,)C(xu . . . , x,_1( *,, x/+1, . . . , xK)].
The following lemma is the key to establishing our main result.
LEMMA 1:
K _
C(0)
n mU, Xi)[mU, 0)}-
C(x) >
for all states x € (Z+U }0))A such that
(i) /€/=#> x, ^ 0
(ii) i <Z I =^> x, = 0
(iii) m (/, Xj) < 1, / € J.
PROOF: The proof is by means of an induction on K. The lemma clearly holds when
K=\ since C(xj) K(\)m(l, xj and C(0) KiX)m{\, 0). We assume that the
lemma holds for an arbitrary problem with K — r — 1 and demonstrate its validity when K — r.
Hence, we consider a problem with rjobs where the position at time 0 is that no jobs have
been completed and job i has been processed for x, units of time where m (/', x,) < 1 and
x, ■ > 0 =£> i € /. Let Sbe an optimal strategy for this problem.
292 K. D. GLAZEBROOK
Suppose that at time 0, 5 chooses to process job 1 (€ /), then
(4) C(x) = ap{Px = x, + l|/>, > x,){- KiX) + C(*,, x2 xr)}
+ ap(Pi > x, + 1|P, > x,)C(x, + 1, x2, ... , xr)
= ap{Px = x, + \\PX > xx){m{\, x,)P {- K(\)m(\, x,)
+ m{\, x{)C(*i, x2 xr)}
+ ap(P] > x, + 1|P, > x,)/m(1, x, + l){m(l, x,)}"1
[C(x, + 1, x2 xXUiHmfUi + Dir1.
Now by our inductive hypothesis
- K(l)m(\, x,) + m(l, x,)C(*,, x2 xr)
(5)
(6)
>
>
- K(\)m(\, x,) + m(\.
*i)
n m(/, *,){/*(/, O)}"1
;=2
C(*i, 0, .
.. , 0)
I! m(i, x,){m(i, 0)}-'
[- K(l)m(\, 0) + m(l, 0)C(*lt 0, ..
, 0)]
fl mU, x,){m(i, O)}"1
C(0),
(5) following from Condition 1 and (6) from the fact that the expression in the square brackets
in (5) is the expected total cost incurred by the DSMNP strategy which first processes job 1
(€ /) to completion and which after that first completion, processes according to an optimal
permutation for the jobs J - 1 1 ) .
We also have that
(7) ap(P\=x^ + l\P\> xOlmil.Xi))'*
+ ap(P\ > X\ + \\P\ >x,)w(l, x, + l){w(l, x,)}~'= 1
and so, from (4), (6) and (7), in order to establish that
(8)
C(x) >
J] mU, x,)[m(i, 0))
/= i
i
C(0).
it is sufficient to demonstrate that we must either have
(9) p(P\ > x, + \\P] > x,)= 0
or that
C(x, + 1, x2 x>(U,)(ffl(l,X| + l))"1^
That is, that
(10) C(x, + l.x2 xr) ^/w(l.x, + l){/n(l, 0)}-'
!!"»(/■ x(-) {/>»(/. 0)}
i= i
Y[mU.xMm(i,0)Y
i=2
-1
C(0).
C(0).
To summarize, in order to establish the desired inequality (8) for state (x,, x2 x,)
it is sufficient to establish the corresponding inequality for state (xj 4- 1, x2 x,) this latter
STRATEGIES IN STOCHASTIC SCHEDULING 293
state being the result at time t = 1 of applying optimal strategy S to the process at time / = 0,
given that no job completion occurs before time t = 1. Should a job completion occur (which
will be job 1) before / = 1 with probability 1 then inequality (8) is satisfied.
We define N* as follows:
N* = inf {N; with probability one the application of optimal strategy S during [0, N)
results in at least one job completion, the initial state being x}.
We further define x(N), 0 < N < N*, to be the state resulting at time / = N from the
application of optimal strategy S to the process from time t = 0 when the initial state is x,
given that no job completion occurs during [0, N). For example, if N* ^ 1 then
x(\) = Oc, + \, x2, ... , xr).
By repetition of the argument in the paragraph following (10) it is clear that in order to
establish (8) it is sufficient to demonstrate that we must have either (i) or (ii).
(i) A/* < oo.
In this case, it is not difficult to show we must have
p[P, > x,(N* - 1) + l\P, > Xj(N* - 1)}= 0
where j is the job chosen by 5 for processing during [N* - 1, A/*) assuming that no job has
been completed prior to N* - 1. Hence, referring back to (9), in the case N* < °°, (8) is
established and the induction goes through.
(ii) N* = oo and
(11) C{x(N)} >
for some N € Z+ U {0}.
II m(i, x,(N)){m(i, 0}"
C(0)
Hence, we now assume that N* = °° (that is, that we cannot be certain of a job comple-
tion under Sin any particular finite time interval) and consider two cases.
CASE 1: x(N) has a single positive component (x,(AO, say) for all N € Z+ U {0}.
When this is so we have that
(12) C(x) = - K(l)m{l, x,) + mil, x,)C{xx *,_,, */( x,+l, ... , xr)
= - K(l)m(l, x,) + mil, x,)C(0 *,, ... 0)
(13) > mil, x,){mil, 0)}-'C(0)
[ mil, x,){mii, 0)}-
C(0),
as required, (12) and (13) following since x, = 0, / ^ /.
CASE 2: xiN) has at least two positive components for all N ^ N, say. When this is
the case it follows from Conditions 1 and 2 that
294
(14)
K. D. GLAZEBROOK
lim [- K(i)m(i, x,(N))]
> lim
N—°°
- K(i)m(i, 0)
11 «0', x,(N)){m(j, 0)]
1 ^ /< r,
and from the inductive hypothesis that
(15) lim in{[m(i, x,(N))C{X](N) *, xr(N))}
^ lim (mU, 0)
,V^oo
n m(J, x,(N)){m(J, 0)]
C(0 * 0)), i € /.
It follows from (14) and (15) that
(16)
C'A lim inf [- K(i)m(i, x,(N))
V— oo
+ m(i, x,(AO)CU,(AO,..
, xr(iV)}]
> lim
,V— oo
-1
^ lim
11 m(J. x,(N)){m(J. 0))
+ mU. 0)C(0 *
|1 '"(.A x,(iV))|m(/, 0)}
{- K(i)m(i, 0)
... 0)}
C(0)
/ 6 /.
Let A7 € Z+ ande > 0 be such that for /V ^W
(17) - K(i)m(i, x,(A0) + m(i, jc,(A0)C{x,(A0 * xr(N)}
> C - e , / € /.
and
(18) m(i, x,(N + s))[m(i, x,(N)))'] ^ (1 +e)~\ s € Z+ U }0), / 6 /.
We shall now demonstrate that for /V ^ N
(19)
C{x(N)} |min C ' - e | (1 + e)'
and, hence, that
(20)
lim inf [C[x(N))] > min C
yV— oo /€/
Having established (20) it will then follow from (16) that
lim inf [C{x(N)}) > lim
V — oo \ — oo
n mU, x,(N))[m(i, 0)]
i=i
C(0)
from which follows the existence of an N € Z+ U {0} for which (11) holds. This established,
the induction will go through and the lemma follows.
We now proceed to demonstrate (19). We consider a problem where at / = 0 no job is
complete and that the state of the process is x(N), N ^ N. Suppose that optimal strategy S
indicates that at time / = 0 (= /0) task j\ should be processed until time / = t](> r0) or until
j\ is completed, whichever occurs sooner. At time / = t\, if j\ has not been completed,
optimal strategy 5 indicates that job j2 (^ j\) should be processed until time t = t2 (> t\) or
STRATEGIES IN STOCHASTIC SCHEDULING
295
until j2 is completed, whichever occurs sooner, and so on. Under the assumption that no job is
completed before time t = t„-\, S indicates that job j„ (^ jn-X) should be processed until time
/ = tn (>J„-x) or until j„ is completed, whichever occurs sooner, 1 < n < °°. It is clear that
for TV > TV
C{x(N)) = £ fl'--' Up{P, > x,(N + ?„_,) |/> > x,(N)}
n=0
1=1
x Z asp[Pln = Xj(N + r„_, + s)\Pj > xJn(N + *„_,)}[- K{jn)
+ C{x,(yV + ?„_,) *j xr(N + r„_,)}]
which, by (18), is
> (i +€)r z ri «(i x,(n + ?„_,))
«=o 1 1 1=1
{«(/, x/(AO)}_,]/>{/>, > X,(/V + *„_!)!/», > x,(/V))]fl'"-'
f — fB_]
x £ asp{P = x (N + ?„_, +s) |/> ■> x, (N + ?„_,)}
S=\
[m(jn, xjn (N + *„_!))}-'
x [- K(j„)m(j„, xJn(N + /„_,))
+ m(/„, x,n(iV + r„_,))C{X,(/V + ?„_,), ... , *,n xr(N + ?„_,)}]];
which, by (17), is
(21)
min C'-e T
) „=o
tl w(/. X/(;v + r„_,)
i=]
(mO, ^OV))}"1/^ > x,(N + t^0\P, > x,(A0}]
xfl--' X **/>{/> ,- x,(N + tH.x + s)\P, > xln(N + /„_,)}
5=1
[m{jn, x/n(N + r,,.,))}-1)
= (1 + e)' (min C'-e],
since the infinite sum in (21) can be shown to be one (the proof is based on (7)). We have
thus established (19) and hence the induction goes through and the lemma follows.
THEOREM 1: There is a DSMNP strategy which is optimal.
PROOF: We may take x, = 0, / € J, in Lemma 1 in which case we obtain that
C(0) > C(0).
Theorem 1 follows immediately.
296 K. D. GLAZEBROOK
3. EXTENSIONS AND COMMENTS
(3.1) Weak Conditions
Theorem 1 continues to hold when the strict inequalities in Conditions 1 and 2 are
replaced by weak ones as follows:
CONDITION 1': m(i, x) ^ m(i, 0), x € Z+, i 6 J\
CONDITION 2': lim m(i, x) exists, i € J
The proof combines the results in Section 2 with a truncation argument of a kind which will be
used in Section 4.
(3.2) Linear Costs
It is frequently the case (see, for example, Glazebrook [3]) that results for problems with
linear costs may be deduced from equivalent results for problems with discounted costs by
means of arguments which involve allowing the discount rate to tend to one. Suppose we con-
sider the problem outlined in Section 1 with costs given by (2). It may be deduced from the
results in the previous section (together with paragraph (3.1)) that under the conditions:
CONDITION 1": nii, x) < nii, 0), x € Z+, / € J\
CONDITION 2": lim n(i, x) exists, / € J, where
piP, > x + 1) > 0-* nU, x)- EiP, - x\P, > x + 1)
p(P, ^ x + 1) > 0=£> nii, x) = 0
there exists an optimal strategy which is DSMNP. This is a generalization of a result due to
Glazebrook and Gittins [4].
(3.3) Continuous Time Analogues
For simplicity our discussion is restricted to discrete time problems. Continuous time
analogues of the main results may be obtained by means of delicate limiting arguments, consid-
ering optimal strategies for appropriately chosen sequences of discrete time problems, allowing
the discrete time quantum to tend to zero.
(3.4) Algorithm Selection
Once we have established that a problem has an optimal strategy which is DSMNP, the
question arises of which permutation (or permutations) determines this optimal strategy. An
algorithm which generates the appropriate permutation for discounted costs (1) may be found
in Glazebrook and Gittins [4]; an algorithm for the linear costs case (2) is to be found in Sid-
ney [10].
STRATEGIES IN STOCHASTIC SCHEDULING 297
4. THE EVALUATION OF NONPREEMPTIVE STRATEGIES
Conditions 1(1', 1") and 2(2', 2"), though they take us much further than the monotoni-
city requirements of previous work, do limit the range of direct application of the material in
Section 2. The main limitation is in the insistence that jobs should always be at least as promis-
ing (i.e., always have at least as low an expected remaining cost) as they are initially. However,
it turns out that the results of Section 2, though limited in this way in their direct application,
help us in the important task of evaluating how well an optimal DSMNP strategy performs rela-
tive to an optimal strategy in a large class of problems of practical interest.
As was implied in the introduction, even if a stochastic job cannot be assumed always to
be at least as promising as it is initially then in many practical contexts such an assumption can
at least be valid for some initial phase of the job's development. For some examples of this,
see Nash [7] whose interest is in modeling research projects and Singh and Billinton [11] who
commend the lognormal distribution as a good model for repair times. Such considerations
motivate the following definitions:
DEFINITION 1: Job i is said to be initially improving for the discounted costs problem if
m(i, 1) ^ m(i, 0) and if lim m(i, x) exists.
X—°o
DEFINITION 2: Job /' is said to be initially improving for the linear costs problem if
n(i, 1) ^ n(i, 0) and if lim n(i, x) exists.
(4.1) Discounted costs
Throughout this subsection we shall assume that all jobs in J are initially improving for
the discounted costs problem. We shall also assume economic criterion (1).
We define
(22) Ti= sup {/; m(i, x) > m(i, 0),0 ^ x < /}, / 6 J.
t€Z+
We further define the random variable P* to be the processing time P, truncated at
7] + 1. Corresponding to P*\s the function m*(i, .). The following lemma is easy to establish.
LEMMA 2:
(i) m*(i, x) > m*U, 0), .v <E Z+, / 6 J.
(ii) lim m*(i, x) exists, /' € J.
Hence, the truncated processing time P* satisfies Conditions 1' and 2' of paragraph (3.1).
Now, the main idea of this section is as follows: suppose that for each job / € 7, T, is large
(which in many practical problems it will be); then the total expected cost incurred by an
optimal strategy will be close to the total expected cost incurred by an optimal strategy for the
equivalent problem with the processing time [P\, P2, ■■■ , Pk\ replaced by the truncated pro-
cessing times {P*, P*, ... , Pk). However, from Theorem 1 and Lemma 2 this latter problem
has an optimal strategy which is DSMNP. These considerations lead us to expect an optimal
DSMNP strategy to perform well relative to an optimal strategy. Theorem 2 aims to quantify
these ideas.
298 K. D. GLAZEBROOK
THEOREM 2:
(C(0)- C(0)}{C(0)}-' <
]1 m*(i, 0){m(i, 0)]
PROOF: Let an optimal DSMNP strategy for the problem with processing times
[Pu P2, ■■■ , Pk\ replaced by truncated times [P*, P* P*K} be given by the permutation
{a(l)( a (2), ... , a(K)}. By Theorem 1 and Lemma 2 this strategy is optimal for that prob-
lem in the class of all feasible strategies. Let C*(0) be the expected total cost incurred by the
application of this strategy to the problem with the truncated processing times and let C(0) be
the expected total cost incurred by the application of this same permutation to the original
problem with nontruncated processing times. It is clear that
Hence,
0 > C(0) ^ C(0) > C(0) > C*(0).
(c(0) - c(o)Hc(0)}-' < (c*(0) - c(o)}{c(o)}
-1
= £- KM')1
,= 1
n m*(a(j), 0)- ft m(fx(j), 0)
/- 1 j= i
t,~ K\pc(i)) ]1 m(a(j), 0)
(=i
^ m «*(«('). 0)- l[m(a(i). 0)
I /= 1 I- 1
II m(o(/). 0)
-l
f[ m*U, 0){m(/, 0)}"
- 1,
as required.
(4.2) Linear costs
Throughout this subsection we shall assume that all jobs in J are initially improving for
the linear costs problem. Costs C(0) and C(0) are as in (4.1) except that now they refer to
economic criterion (2).
We define as before
(23)
S,= sup {t\ n(i, x) ^ n(i, 0), 0 ^ x ^ t)
and thus obtain function n*(i, .) as in (4.1). This function is found to satisfy Conditions 1"
and 2" and so we have Theorem 3.
THEOREM 3:
|C(0) - C(0)}{C(0)}-' < max [{/»(/, 0) - n*(i, 0)}{n*(i, 0)) '].
1 < / ^ K
PROOF: The proof is similar to Theorem 2.
We deduce from Theorems 2 and 3 that when dealing with collections of initially improv-
ing jobs whose associated values of T, and S, are large we lose little by restricting our attention
STRATEGIES IN STOCHASTIC SCHEDULING
299
to DSMNP strategies. Note too, that in any given problem it may be that we can truncate at times con-
siderably larger than Tt + \ or 5, + 1 and still have functions m*(i, .) or «*(/', .) satisfying the
appropriate conditions. When this is the case it may be possible to improve the bounds given in
Theorems 2 and 3.
Note further that Theorems 2 and 3 also hold in continuous time. The modifications
required are that in the definitions of Tt and S, in (22) and (23) respectively the suprema
should be taken over R+, the nonnegative real numbers, and that to obtain P* in both cases,
truncations are taken at 7] and S, respectively. We also need to modify Definitions 1 and 2 in
the obvious way.
5. EXAMPLE
For simplicity, we consider an example in continuous time with linear costs as in (2).
There are five jobs and so J = {1,2,3,4,5} with predence relation R = {(1,2), (1,5), (2,3),
(5,3)}. It is not difficult to see that there are ten feasible DSMNP strategies for J. The distri-
bution of Pj is summarized by its hazard rate X,(.) which is assumed to have the form
(24)
kj(x) =
\i„ 0 < x < TXi,
\2i, TXl ^ x < Tu + T2i,
k3i, Tu + T2l < x.
i= 1,2,3,4,5.
The important details for the five jobs are summarized in Table 1. It is easy to show, by appli-
cation of the algorithm due to Sidney [10] that the optimal permutation is (4,1,5,2,3) with
associated expected cost C(0) = 31.089.
TABLE 1
Job (i)
K(i)
M,
k2i
k3i
Tu
T2i
n(i, 0)
1
1
1
3
2
1
1
0.758
2
2
1
3
1.5
1
2
0.817
3
3
2
5
2.5
2
4
0.495
4
4
2
4
3
2
1
0.495
5
5
1
2
1
3
3
0.975
It is also not difficult to demonstrate that, with processing time distributions given accord-
ing to (24) that
(25) A 2, > M/ and \3i > \f
where
(xrr1=a1/)-1{l-exp(-X1,T1,)} + a2;rl{exp(-\1/T1;)-exp(-X1;T1;~X2/T2,)}
x {l-exp(-\1/T1,-A2;T2;)}-1
are sufficient to ensure that
n(i, x) ^ n(i, 0), x € R+,
and the existence of
lim n(i, x).
300 K. D. GLAZEBROOK
Jobs 1, 2, 3 and 4 all satisfy (25) but job 5 does not. Indeed,
n(5, x)= 1 > n(5, 0), x > 6.
However, job 5 is initially improving in the sense that the (right-hand) derivative of n(5, x) at
x = 0 is negative, and so the theory of Section 4 applies. In fact, the value S5 can be shown to
be 5.975 and the continuous-time version of Theorem 3 applied to this case yields
(C(0) - C(0)}{C(0)}-1 ^ {h(5, 0) - n*(5, 0)}{«*(5, 0)}-'= 1.30 x 10"4,
whereupon we obtain, that
31.085 < C(0) ^ 31.089.
Evidently, then, very little is lost in this case by restricting attention to permutations of /
REFERENCES
[1] Garey, M.R., "Optimal Task Sequencing with Precedence Constraints," Discrete Mathemat-
ics, 4, 37-56 (1973).
[2] Glazebrook, K.D., "Stochastic Scheduling with Order Constraints," International Journal of
Systems Science, 7, 657-666 (1976).
[3] Glazebrook, K.D. "On Stochastic Scheduling with Precedence Relations and Switching
Costs," Journal of Applied Probability, 17, 1016-1024 (1980).
[4] Glazebrook, K.D. and J.C. Gittins, "On Single-Machine Scheduling with Precedence Rela-
tions and Linear on Discounted Costs," Operations Research, 29, (1981, to appear).
[5] Meilijson, I. and G. Weiss, "Multiple Feedback at a Single Server Station," Stochastic
Processes and their Applications, 5, 195-205 (1977).
[6] Monma, C.L. and J.B. Sidney, "Sequencing with Series-Parallel Precedence Constraints,"
(submitted for publication).
[7] Nash, P., "Optimal Allocation of Resources between Research Projects," Ph.D. Thesis,
Cambridge University, Cambridge, England (1973).
[8] Ross, S.M., Applied Probability Models with Optimization Applications, (Holden-Day, San
Francisco, Calif., 1970).
[9] Rothkopf, M.E., "Scheduling Independent Tasks on Parallel Processors," Management Sci-
ence, 12, 437-447 (1966).
[10] Sidney, J.B., "Decomposition Algorithms for Single Machine Sequencing with Precedence
Relations and Deferral Costs," Operations Research, 23, 283-298 (1975).
[11] Singh, C. and R. Billinton, System Reliability-Modelling and Evaluation, (Hutchinson & Co.,
London, England, 1977).
POSTOPTIMALITY ANALYSIS IN NONLINEAR INTEGER
PROGRAMMING: THE RIGHT-HAND SIDE CASE
Mary W. Cooper
Department of Operations Research and
Engineering Management
Southern Methodist University
Dallas, Texas
ABSTRACT
An algorithm is presented to gain postoptimality data about the family of
nonlinear pure integer programming problems in which the objective function
and constraints remain the same except for changes in the right-hand side of
the constraints It is possible to solve such families of problems simultaneously
to give a global optimum for each problem in the family, with additional prob-
lems solved in under 2 CPU seconds. This represents a small fraction of the
time necessary to solve each problem individually.
1. INTRODUCTION
Recently efforts have been made to extend the ideas of postoptimal analysis and
parametric analysis which are widely used in linear programming to 0-1 integer programming
and general integer programming. A review of these efforts is given by Geoffrion and Nauss
[4]. They cite work on the 0-1 problem by G. Roodman [13], and an extension of that work by
Piper and Zoltners [11]. Roodman [12] and Marsten and Morin [8] have looked at the same
topic using branch and bound. These and other authors are cited in [4]. Bailey and Gillett [1]
have recently used cutting planes in parametric integer programming. The present paper differs
from these efforts in considering postoptimal right-hand side analysis for a different problem:
the pure integer nonlinear programming problem with separable objective function and con-
straints. Our purpose is to modify an algorithm which has been previously described [3] so that
it simultaneously finds optimal solutions for a family of problems of the type described above
which differ only in the right-hand side vector of the constraints. (This family is analagous to
Geoffrion and Nauss' family Pw in their discussion of postoptimality analysis for the linear
integer case).
2. APPLICATIONS
One of the most general formulations to which this algorithm applies is the separable non-
linear knapsack problem. It has numerous application areas in allocation of resources, cutting
stock problems and capital budgeting [7], [9], [10], [5], [6]. In addition it has applications for
solving subproblems in many integer programming algorithms [14], [2], [15]. The importance
of the work in this report which gives postoptimality data for this problem can be argued in a
way analagous to the case for linear programming. Additional information about the value of
301
302 M. W. COOPER
changes in resources, is usually worth a minor amount of additional computation. Often right-
hand side values represent estimates, and information about the effect of right-hand side
changes on the optimal solution represents a crude determination of the effect of estimating a
variable by its expected value.
3. THE PROBLEM AND METHOD
Let us first characterize the problems we solve, and second, briefly review the elements of
the algorithm to be modified. After these sections, the algorithm is extended to solve the fam-
ily of problems which differ only in the right-hand side vector.
Let us use the following notation to formulate the problem (P).
n
(1) Maximize z = £ /,(*,) subject to
7=1
n
(2) £ hij(xj) ^ bt, i = 1,2 m, and x, € Ip for j = 1 n.
7=1
Additional restrictions on the functions are
(1) fj:Ip—'Rp,j= 1 n, and they satisfy a sufficient condition for dynamic pro-
gramming.
(2) hy : Ip — » Rp, j = 1 n, and /' = 1, .... m and are nondecreasing in x-r
(3) the region described by the constraints is nonempty, contains at least one integer
point, and is bounded.
Our previous algorithm [3] is a top-down enumerative method for solving this problem in
which the constraints are used to eliminate infeasible partial solutions and their completions. In
this paper we require the additional restriction described above in condition (2), although the
paper cited in [3] treats a more general nonseparable form of the constraints. Let us describe
the solution process for the pure integer nonlinear separable programming problem given in (1)
and (2).
Step 1: Find upper bounds on Xj, j — 1, . . . , n and z0 over the constraints in set (2).
Step 2: Solve the following dynamic programming problem:
n
(3) Maximize Z = £ //(■*/)
7=1
n
Subject to £ fj{xj) < z0.
7=1
This single dynamic programming problem can be used to identify lattice points on the hyper-
surface
n
X fj(Xj) = Z0
7=1
and on every hypersurface
POSTOPTIMALITY ANALYSIS IN NONLINEAR INTEGER PROGRAMMING 303
(4) £ fj(xj) = z, 0 < z ^ z0.
7=1
Step 3: We use the dynamic programming solution table to generate both a sequence of
decreasing values of z which correspond to hypersurface levels containing integer
points and also to generate all lattice points on that particular hypersurface. For
details of the method, see [3].
Step 4: The constraints (2) of the original problem are used to check for feasibility. The
argument is simply, if we look at all hypersurfaces (4) in decreasing order of z,
then the first feasible point with respect to the constraints (4) will be optimal.
Actually the feasibility of the solutions is checked at the partial solution stage. For a
given z, say zk, we generate the components of the lattice point in the order x*, x*-\, . . . , x*.
After x* is generated, the vector corresponding to the remaining resource levels, that is,
b'= b - anx*
is checked for any negative components. If none are found, this partial solution is still a candi-
date for a feasible solution. Otherwise it is eliminated before any other components x*_i,
x*-2, . ■ ■ , x* are generated from the dynamic programming tables, since the final solution is
infeasible no matter what the remaining components are. Hence, solutions are eliminated from
consideration as quickly as possible.
4. ADDITIONAL CALCULATIONS TO DETERMINE OPTIMAL SOLUTIONS FOR
CERTAIN MEMBERS OF THE Pe FAMILY
Let us assume that we want to find optimal solutions to the following problem Pa
n
Maximize z = £ /}(*/)
n
Subject to £ hjj(xj) < b: + 6 rt i = 1, . . . , m.
7=1
Xj 6 Ip for j — 1 , . . . , n.
0 = 0O < 0i < ... < 0/ = 1
r, > 0, /= 1, ... , m.
Then Step 4 must be changed toinclude additional tests for feasibility for each of the
right-hand side vectors, b0 = b, bt = b + 9{ 7, b2= b +G27, b3= b + 037, . . . , b, = b + r.
Note that if 0 < 0! < 02 . . . < 1, the following relationship between the right-hand side values
exists—
bt < bn = biQ + 9xr, < bi2 = biQ +92rl < ... < bn= b, + ri
for / = 1 m.
Let us assume that we are testing the feasibility of a partial solution with constraint i. Then if
feasibility is tested for bih bn-X, ... , b}, b0, if any constraint is violated whose /th constraint
has right-hand side value bip, then for the problems whose right-hand side values are bitP-\,
bjp_2 bn, the current partial solution will also be infeasible. This is the order of calcula-
tion that has been implemented in a computer program. It is also possible to describe an algo-
rithm for solving a set of problems whose right-hand side vectors are not related as those are in
304 M. W. COOPER
the Pft family which decrease in every component. For two problems with arbitrary^ and
differing right-hand side vectors t>\ and t>2, then there may be no method of ordering the b vec-
tors so that for every row /, bn < ba. Hence, a less efficient algorithm could be implemented
in which every bi{ must be checked, even if an indication of infeasibility is given for a previous
bu+x. The reason is obvious: for arbitrary components no ordering can guarantee that
b, i < bjj+] for every constraint, hence, the /th constraint may not be violated if its right hand
side is bu.
A flow chart of the order of the calculations for implementation of the simultaneous solu-
tion of a family of problems differing in the right-hand side is given below. We assume that we
have generated an upper bound z0 on the objective function in some way, and that we are con-
sidering a partial solution for some hypersurface with functional value zk < z0. The assump-
tion is clearly that for all hypersurfaces with intermediate functional values either
(a) they contain no integer points (we do not explicitly consider these), or
(b) they contain no feasible integer points.
At each stage in generating a new component of an integer point from the dynamic program-
ming tables a test for feasibility is made with the new x* and components in the partial solution
already obtained. Hence, the flow chart of this part of the algorithm assures that the sequence
of functional hypersurface values with integer points has been identified and put in strictly des-
cending order: zQ > Z\ > . . . > zk > . . . . The right-hand side vectors under consideration can
be written as
bp = b +9p7, and 0 < 0, < 92 < 9p < 0,= 1.
The program considers the right-hand side vectors in the order bh 6/_j, ... , b\, b0, so that any
partial integer solution which is infeasible for bp is also infeasible for all previous right hand-
side vectors. The logic is given in the following diagram (Figure 1).
A careful analysis of the program logic will show that many problems of the family PH can
be solved using the solution table from a single dynamic programming problem. We would
expect a considerable saving over the time for solving each problem in the family separately for
this reason. In addition, the fathoming or discarding of integer points at the partial solution
stage can be done for several problems at a time.
5. COMPUTATIONAL DATA
Seven different basic families PH have been solved on the CDC CYBER 70, Model 72, a
moderate speed computer. The results are given in Table 1. m is the number of constraints, n
is the number of variables, k is a bound on xr Problems are created with randomly generated
coefficients. The functions of fjixj) are cubic polynomials, so a problem with 12 variables
might have as many as 36 terms in the objective function. Constraints of the form
n
£ OjjXj < bit /= 1, ... , m
/=i
are used with the restriction that atJ ^ 0, b, > 0. For each member of the Pq family, the new
right-hand side vector is created by subtracting 5 from each component of b. A time of .00
indicates that the current optimal solution also solves the next problem in the family which has
a smaller value in each component of b. Note that other schemes of obtaining members of P»
can be easily implemented.
POSTOPTIMALITY ANALYSIS IN NONLINEAR INTEGER PROGRAMMING
305
^
Enter with partial
solution and objective
function value z.
Z = Zk+1 < Zk
Get Smaller
right- hand side
None smaller
±kL
Stop
Test
Solution
with new
1R.H.S. for
easibility ,
Not feasible
v
©
Figure 1. Flow chart
306
M. W. COOPER
TABLE 1
Time (CPU Seconds)
m
n
k
Base Problem (b\)
(seconds)
b2
b)
b4
bs
3
12
2
26.43
.00
.55
1.12
.56
4
10
2
13.27
.00
.00
1.38
.44
4
10
2
30.36
.00
.00
.00
1.64
4
15
2
36.26
.00
.00
.00
.00
4
15
2
84.49
1.33
.00
3.87
1.09
4
15
2
32.98
.00
.00
.00
.00
4
15
2
25.70
.00
.00
.00
.66
In each case, although individual problems are between 10-80 CPU seconds, after the
base problems are solved, other problems in the same family are solved in under 2 seconds.
6. ACKNOWLEDGMENT
The author would like to acknowledge the valuable comments of T.L. Morin on the topic
of this paper.
REFERENCES
[1] Bailey, M.G. and B.E. Gillett, "Parametric Integer Programming Using Cutting Planes,"
unpublished paper, ORSA/TIMS Meeting, Los Angeles (December 1978).
[2] Bradley, G., "Transformation of Integer Programs to Knapsack Functions," Discrete
Mathematics, /, 29-45 (1971).
[3] Cooper, M.W., "The Use of Dynamic Programming Methodology for the Solution of a
Class of Nonlinear Programming Problems," Naval Research Logistics Quarterly, 27,
No. 1 (1980).
[4] Geoffrion, A.M. and R. Nauss, "Parametric and Postoptimality Analysis in Integer Linear
Programming," Management Science, 23, No. 5 (1977).
[5] Gilmore, P.C. and RE. Gomory, "Multi-Stage Cutting Stock Problems of Two or More
Dimensions," Operations Research, 13, 94-120 (1965).
[6] Gilmore, P.C. and R.E. Gomory, "The Theory and Computation of Knapsack Functions,"
Operations Research, 14, 1045-1074 (1966).
[7] Lorie, J.H. and L.J. Savage, "Three Problems in Rationing Capital," Journal of Business,
28, 229-239 (1955).
[8] Marsten, R.E. and T.L. Morin, "Parametric Integer Programming: The Right-Hand Side
Case," Discrete Mathematics, /, 375-390 (1977).
[9] Nemhauser, G.L. and Z. Ullman, "Discrete Dynamic Programming and Captital Alloca-
tions," Management Science 75, 801-810 (1969).
[10] Peterson, C.C., "Computational Experience with Variants of the Belos Algorithm Applied
to the Selection of R & D Projects," Management Science, 13, 736-750 (1967).
[11] Piper, C.J. and A. A. Zoltners, "Some Easy Postoptimality Analysis for Zero-One Program-
ming," Management Science, 22, No. 7 (1976).
[12] Roodman, G.M., "Postoptimality Analysis in Integer Programming by Implicit Enumera-
tion: The Mixed Integer Case," The Amos Tuck School of Business Administration,
Dartmouth College (October 1973).
[13] Roodman, G.M., "Postoptimality Analysis in Zero-One Programming by Implicit Enumera-
tion," Naval Research Logistics Quarterly, 19, No. 3 (1972).
POSTOPTIMALITY ANALYSIS IN NONLINEAR INTEGER PROGRAMMING 307
[14] Salkin, H.H. and C.A. De Kluyver, "The Knapsack Problem: A Survey," Naval Research
Logistics Quarterly, 22, 127-144 (1975).
[15] Shapiro, J.V. and H.M. Wagner, "A Finite Renewal Algorithm for the Knapsack and
Turnpike Models," Operations Research, 15, 319-341 (1967).
AN EFFICIENT ALGORITHM FOR
THE LOCATION-ALLOCATION PROBLEM
WITH RECTANGULAR REGIONS
Ann S. Marucheck
Oklahoma City University
Oklahoma City, Oklahoma
Adel A. Aly
School of Industrial Engineering
University of Oklahoma
Norman, Oklahoma
ABSTRACT
The location-allocation problem for existing facilities uniformly distributed
over rectangular regions is treated for the case where the rectilinear norm is
used. The new facilities are to be located such that the expected total weighted
distance is minimized. Properties of the problem are discussed. A branch and
bound algorithm is developed for the exact solution of the problem. Computa-
tional results are given for different sized problems.
1. INTRODUCTION
All previous studies of the location-allocation (L—A) problem have used the assumption
that the location of customers of existing facilities were deterministic points. The multifacility
location problem involves the location of one or more new facilities relative to several existing
facilities in order to minimize the sum of the weighted distances among the facilities. Previous
work [1,2,16] with this problem has shown that in the urban setting, potential location of custo-
mers or existing facilities may be more accurately represented as random points uniformly dis-
tributed over rectangular regions. Since the L—A problem is a generalized version of the mul-
tifacility location problem, the principal of using rectangular regions to represent existing facili-
ties instead of aggregate points would be appropriate in modeling the L—A problem.
A common approach to handling the location problem with rectangular regions is to
represent each region by its centroid and to solve the resulting problem as a deterministic
model. Although this method is computationally easier, it has been shown [3] that the
solutions's proximity to optimality is metric dependent. Location problems with Euclidean dis-
tance metric are relatively insensitive to a relaxation of the probabilistic assumptions. In other
words, using the centroid approach for probabilistic location problems with Euclidean distance
metric yields a near optimal solution. However, the tradeoffs in considering the deterministic
(centroid) version of the rectilinear metric location problem are greater [1]. Consequently, in
considering probabilistic location formulations using the rectilinear metric it is necessary to
develop solution techniques other than the deterministic ones.
309
310 AS. MARUCHECK AND A. A. ALY
Often the solution techniques for the L—A problem involve the use of a facility location
algorithm to generate and evaluate allocation schemes. Cooper [6] and Kuenne and Soland [9]
both indicate that finding the optimal allocation scheme is the most critical task in solving the
L—A problem. Thus, determination of the optimal allocation scheme is only as reliable as the
facility location techniques employed.
The purpose of this research effort is to develop and test an exact solution technique for
the L—A problem among rectangular regions with a rectilinear metric.
2. FORMULATIONS
The general location-allocation model among rectangular regions is formulated as follows.
ftl /* /»
(P) minimize Y*Y* ) ) Zu»i\Xj ~ R,\i 0(R,)dR,
7-1 i=l R, P
n
subject to: £ zu = 1 for all i
J=x 2U = 0, 1 for all i and 7
where: n = number of new facilities
m = number of existing facilities
Xj — (xj, yf), coordinate location of new facility y"
R, = existing rectangular region i
9(Rj) = bivariate probability density function over R,
Wj = interaction between region /and the new facility it will be allocated to
1, if existing facility / is allocated to new facility j
0, otherwise
lp = the type of norm used. When p = 1,2, and °°, the metric becomes
rectilinear, Euclidean, and Chebyshev distances respectively.
The particular problem to be emphasized in this paper is the location-allocation problem
among rectangular regions with bivariate uniform distributions.
This may be expressed as,
(PO minimize £ £ -^ J J \ XJ - R,\t da.db,
n
subject to: £ Zy ■ = 1 / = 1 m
J=x Zij = 0, 1 for all / andy
where: (a,, b,) = general coordinate location in region R,
A, = area of region R,
and n, m, w,, Xj, /?,- and z,7 are as defined in (P).
Note that — in (P') is just the bivariate uniform density function over /?,.
In Problems (P) and (P') , the decision variables are the z,/s-reflecting the allocation
aspects of the problem and the A^'s-reflecting the location aspects of the problem.
LOCATION-ALLOCATION PROBLEM 3 1 1
The new facilities have an infinite capacity to serve the existing facilities. Thus, each
existing facility will be allocated to and subsequently interact with only the closest new facility.
It is assumed throughout that the w,'s may represent either deterministic values or
expected values of random variables. Also, the regions must be rectangular, but they may be
overlapping.
3. RELATED WORK
There has been no previous work on the L—A problem among regions. However,
research on the deterministic version of the problem has revealed the complexities and compu-
tational burden involved in the solution of the L—A problem.
In light of the difficulties associated with exact solution of the L—A problem, heuristic
algorithms are often employed. Cooper [5,6,7] developed various heuristic algorithms. Many
of his initial algorithms used the assumption that all existing facilities were equally weighted; he
used these results to develop heuristic for the case when the facilities are not equally weighted.
Learner [10] assumed customers were uniformly distributed over a plane and attempted to allo-
cate them to the new facilities by dividing the plane into hexagonal areas.
Since the heuristics can not guarantee a specific proximity to optimality, exact algorithms
have been developed with an attempt to alleviate the computational burden of the L—A prob-
lem. Most algorithms have concentrated on the Euclidean metric. Bellman [4] was able to
solve very small L—A problems by transforming them into dynamic programming problems
using quasilinearization as the transformation device. Kuenne and Soland [9] used a branch
and bound algorithm to optimally solve the L—A problem with Euclidean, great circle, and rec-
tilinear distance metrics. Ostresh [12] worked on the Kuenne and Soland algorithm in an
attempt to improve the bounding procedure. He did so for the case n = 2 using convexity
results of Wendell and Hurter [15]. Love and Morris [11] considered the L—A problem with
rectilinear norm. Their exact algorithm features a reduction scheme where only possibly
optimal sites for new facilities are considered. Recently, Sherali and Shetty [14] used a cutting
plane algorithm to solve the L—A problem with rectilinear norm.
Although these exact methods can guarantee optimality, there are limitations to the size
of problem that can be solved in terms of computational time. Ostresh [12] reported solving
problems of sizes m = 23, n = 2 and m = 11, n — 4 in respective CPU times of 23.26 sec and
10.28 sec on IBM 360/65. Kuenne and Soland's [9] largest reported problem was m = 15, n —
4 with CPU times for random weights and unit weights, respectively, of 82.7 sec and 54.2 sec
on an IBM 360/91. Sherali and Shetty [14] solved a problem of size m = 35, n = 2 in 23.46
seconds on a CDC 6600. Finally, Love and Morris [11] reported solving a problem of size m
= 35, n = 2 in one hour and 31 minutes of CPU time on a Univac 1110. Thus, computational
burden seems to be a serious problem for exact solution methods.
4. A BRANCH AND BOUND APPROACH
The branch and bound approach developed by Kuenne and Soland [9] offers an optimal
solution to the L—A problem in reasonable computational time. Although Kuenne and Soland
developed a solution for the deterministic problem, some of their results may be generalized
and adapted to the form of the L -A problem considered here. Some of the generalized results
are discussed below.
312 A. S. MARUCHECK AND A. A. ALY
The L—A branch and bound algorithm is based on partitioning the set of all possible solu-
tions to the location-allocation problem on the basis of the allocations of the existing facilities
to the new facilities.
Any subset of solutions, denoted S, can be partitioned into at most n disjoint sets by con-
sidering the total number of ways a previously unallocated existing facility can enter the alloca-
tion scheme. Suppose that in S the allocated existing facilities have been assigned to k new
facilities where k ^ n. An unallocated existing facility is chosen. If k = n, then 5 can be par-
titioned or separated into /; subsets S,, S2, .... Sn where Sj is characterized by the assignment
of the existing facility to new facility j. On the other hand, if k < n, then S may be partitioned
into K + 1 subsets where S}, j = \, 2, .... k is as described above. The subset Sk+] is
characterized by the assignment of the existing facility to a (A + 1 ) th new facility. This
(k + Dth new facility would have only one existing facility allocated to it.
After a node or subset A has been partitioned, a lower bound is computed for each parti-
tion or succeeding node j to help in fathoming the generated nodes. This bound is a lower
bound on the objective function value that would be produced by any allocation scheme con-
taining the allocations that have been made at this node j. The lower bound is a sum of two
values. The first value is the cost of optimally locating the new facilities among the existing
facilities that have been allocated; this is just a multifacility location problem. The second value
is a lower bound on the cost of locating n new facilities among the unassigned existing facilities.
When the mth level is reached a complete allocation scheme has been developed, as each
of the m existing facilities has been allocated to one of the n new facilities.
4.1 The Branching Rule
The branching rule is the criterion used to choose the unallocated existing facility at each
level whose assignment will be considered as the basis for making the partition. Any rule may
be used. For example, an unallocated existing facility could be chosen at random or the /th
existing facility could be chosen as the branching facility at the /th level. However, an approach
based on the properties of the problem may be more useful.
For this problem where the sum of weighted expected distances is to be minimized, the
weighted expected distance from an existing facility to a new facility will be considered as a
branching rule as a generalization of the results of Kuenne and Soland [9]. Considering only
the minimum distance or maximum distance between an existing facility and all new facilities
would disregard the size and variations of the expected distances between the existing facility
and the new facilities.
The weighted expected distance between region / and new facility j is
(1) ^-$b'2 \ra'7{\xJ-al\+\yJ-bl\)da,dbl
where all parameters are defined as in (P) and (P').
This is equivalent to the following expression:
a b
W r '■> W /* '7
(2) 7rrr^-^ + rrrI \yj-A\db,
%- %•>% J b,2- b,
i 'i
LOCATION-ALLOCATION PROBLEM
313
Each expression in the sum may be computed independently. Hence, because of this
separability there is an expected distance with respect to the x-coordinate and another with
respect to the ^-coordinate.
It may be shown that the expected distance from (x, y) to region i defined by
[ai]t ai2] x [bir bj2], where xS'Ca, , a,) and y$(bj , 6,), is equivalent to the rectilinear dis-
( ah + ah K + b^
tance from (x, y) to the midpoint of these intervals
presented in Theorem 1.
2
Another case is
These expected distances are used in both applying the branching rule and evaluating the
objective function.
4.2 Upper and Lower Bounds
4.2.1 Bounds on the Objective Function
The objective function value associated with an arbitrary allocation scheme may serve as
an upper bound. This upper bound may be improved by using a modification of Cooper's alter-
nate location and allocation heuristic [6].
Consider the arbitrary allocation where existing facility / is allocated to new facility j where
Ii (mod n), if j is not divisible by n
n, otherwise.
By this definition existing facility n would be allocated to new facility //, but existing facility
n + 1 would be allocated to new facility 1.
The location problem for this allocation is solved and the objective function value com-
puted. This is an upper bound on the optimal solution value. The upper bound is tested for
improvement by reallocating each existing facility to the new facility whose weighted expected
distance from the former facility is a minimum. After the reallocations are made, the location
problems are again solved and a new objective function value computed. If the new objective
function value is equal to the old objective function value, iterations cease. Otherwise, the
reallocations start again. This heuristic may be iterated until no improvement is made or until a
convergence criterion is met. The best objective function value from this heuristic becomes the
upper bound on the optimal objective function value. The minimum expected cost of serving a
region is established in the next theorem.
THEOREM 1: The minimum expected cost, TM of serving region /from a point within /',
w
is -j- (a, — Qj + />,, — b, ) where region / is defined as [a, , a, ] x [blr b,A.
PROOF: The expected cost of serving region / from (x, y) a point within / is
(3)
fix, y)
wt
(a, ~ x)2 + (a. - x)2 (bh - y)2 + (b, - y):
2(dj- ah)
+
2(b,-bh)
When the partial derivatives of (3) are set to 0, the solution is
% + a bh + b
Gc* y*) =
314 A. S. MARUCHECKAND A. A. ALY
which yields a minimum. Thus,
w
fix*, y*) = -f (al} - ah + b,2 - b,{) = T,.
The lower bound to the objective function may then be found by:
(4) Lb. = £ Tr
1=1
4.2.2 A Lower Bound for Each Node
Computing a lower bound is a two part process. The first part is solving the location
problem for the allocated existing facilities and computing the corresponding new facility. The
second part involves underestimating the expected cost of locating the n new facilities among
the unallocated existing facilities.
In order to develop the second expression, consider two unallocated regions R\ and R2.
Suppose that both are to be served by the same new facility X = (x, y). The expected cost of
serving these two regions is:
(5) fiX) = wxE[\x - fl,| + \y- 6,|] + w2E[\x - a2\ + \y ~ b2\]
where (a,, b,) are random variables representing the points located in region /'. This expression
can be considered the sum of the expected costs of serving the regions along the x-coordinate
and the expected cost of serving the regions along the >'-coordinate. These expressions are
independent and each one-dimensional case may be considered separately.
Notice that when the x-coordinate is considered, then the expected cost is
(6) fix) > min{whw2} iE[\x - a,|] + £[|x - a2\\).
Let a\ and a2 assume any values where a\ < a2 and consider the relative position of x. By the
triangle inequality,
(7) |x - ax\ + \x - a2\ ^ |a, - a2\.
Since a\ and a2 are random variables, then
(8) £[|x - a,|] + E[\x - a2\] > E[\ax - a2\\.
Substituting (8) into (6), a lower bound is produced:
(9) fix) ^ min{wlt w2) E[\a] - a2\}.
Thus, (9) is an appropriate lower bound, where E[\a\ - a2\] represents the expected dis-
tance between regions 1 and 2 along the x-coordinate.
(10) £[|fl|-02l]=f f 2\u-v\dudv.
1 * J a-, J a-,
11
The integral in (10) may be evaluated for three cases. For ease in reading, let a represent
ci\ , b represent ax , c represent a2 , and (/represent a2, (the second interval a2 is underlined).
LOCATION-ALLOCATION PROBLEM
CASE I. a < c < d < b
315
(11)
a2
E[\a] - a2\] =
(a2 + b2)(d - c) - (a + b)(d2 - c2) + j (d3 - c3)
2(b-a)(d- c)
CASE II. a < c < b < d
ai
a\
(12) E[\ai-a2\Y-
(b-c)[(a2 + c2)-(a + b)(b + c)] + j(b3-c3) + (d-b)(b-a)(d-c)
2(b-a)(d-c)
CASE III. a < b < c < d
ai
(13)
E[\a\ - a2\]
id2- c2)(b- a)- (d- c)(b2- a2)
Kb - a)(d- c)
From (13) it can be shown that if the two regions R\ and R2 have nonoverlapping inter-
d + c - b - a
vals, then the expected distance between the two is just
2
Thus, for any two rectangular regions R, and Rn the expression
(14) min{w;) h-/}(£'[|o,- a,\ + \b, - bj\])
can be computed as an underestimate of the expected cost of serving these two regions with the
same new facility (see [6]).
Thus, Equation (14) is the building block for forming lower bounds. If there are p unal-
located existing facilities, then there are \/2p(p — 1) different realizations of Equation (14).
Assume that all the expressions are placed in ascending order and let q, be the /th term in this
progression. Compute Tt for ./ = 1 f, where 7} is defined as in (4), and arrange these
expressions in ascending order. Let /•, be the /th term in this progression.
316
A. S. MARUCHECK AND A. A. ALY
To underestimate the expected cost of allocating /; new facilities among p existing facili-
ties, the various combinations of allocations should be studied. For example, if p < «, then a
new facility should be assigned to each of the p existing facilities. An underestimate of this
cost would be the sum of all p of the /', terms. This would follow since r, represents a
minimum expected cost for serving a region from a point in the region.
Another example is the case where p = n + 4. In this case, there are five possible combi-
nations: four new facilities are allocated two existing facilities, all others are allocated one; one
new facility is allocated three existing facilities, two are allocated two, and the others are allo-
cated one; one new facility is allocated four existing facilities, one is allocated two, and all oth-
ers are allocated one; two new facilities are allocated three existing facilities apiece, and all oth-
ers are allocated one; and finally one new facility is allocated five existing facilities, and all oth-
ers are allocated one.
Table 1 displays all lower bounds for these combinations for different values of p - n.
TABLE 1 — Lower Bounds for Locating n New Facilities Among
p Rectangular Regions
Value of
(p- n)
Lower Bound
0 or less
1
2
i=l
n-1
</i + Z n
mm
mm
n-2
a\ + </2 + Z ri> 1/2(^/l + °2 + </j) + Z ri
i-l
n-3
n-1
U\ + <ll + Qy + Z 'm <l\ + l/2(</2 + </3 + </4> + Z r,,
i=\
y (</, + ... + <jr6)+Z l)
mm
11 A n— 3
<l\ + • ■ • </4 + Z '/• "\ + u2+ 1/2((/3 + </4 + q5) + Z rh
<l\ + 4- (q2 + •■• + <7?) + Z '•/- l/2(«i + • • • + 46) + Z r„
J n-1
— {qx + ... + tf10) + X '',
In— 5 n— 4
qx + . .. + qs + Z '/• ?i + </2 + </3 + l/2((/4 + <y5 + </6) + Z Oi
i=i i-i
</i + </2 + T («3 + . • ■ + </s) + Z r„ q\ + l/2(</2 + . . . + q7) + JT ,-,.
3
;=1
n-2
</, + l/4(^2 + . . + qu) + Z ',• l/2(</i + ff2 + «3) + J. («4 + ■■ • + </9)
/=1
n-2
M-l
+ Z 0. T fal + •■•+ tfis) + Z '<
:_1 J ,_ I
LOCATION-ALLOCATION PROBLEM 317
It is obvious that as p — n becomes larger than five, the number of combinations to be
considered also becomes large. Thus, a general lower bound will be used for values of p — n
greater than five.
THEOREM 2: A general lower bound on locating n new facilities among p rectangular
regions is 1/2 ^ q, where q, is as defined above. The proof follows closely that in [6,9].
This general lower bound is well-suited for the cases when p — /; is large. These cases
will be levels 1, 2 m — 5 of the tree. At these levels, the possibility of fathoming nodes
is not as great as at the other levels. This is because only a few existing facilities have been
allocated, and the partial objective function value used in computing the lower bound will be far
from the optimum. A tight lower bound would then involve considering all possible combina-
tions of the unallocated facilities. To hasten the tree search, the general lower bound is used to
quickly compute the lower bound and move to the next level.
On the other hand, in the last « + 5 levels of the tree, enough facilities have been allo-
cated to identify unprofitable allocation schemes. Here the tighter lower bounds given in Table
1 should be used to fathom as many nodes as possible.
5. THE LOCATION-ALLOCATION BRANCH AND BOUND ALGORITHM (LABB)
In this section the complete branch and bound algorithm for the location-allocation prob-
lem is given.
The input parameters are
N = number of new facilities
M = number of existing regions
x\ (/) and .v2(/) = left and right endpoints, respectively, of region l(R (/)) along .v-axis
y\(I) and^;2(/) = lower and upper endpoints, respectively, along >-axis.
w(l) = interaction cost for region /.
The parameters for computing bounds on the optimum value of the objective function
are:
z = upper bound on optimum
z = lower bound on optimum
FX = current least upper bound on optimum
F = objective function value to be compared with FX
e = stopping criterion for alternate heuristic (e > 0).
The parameters for computing the branching facility are:
L = current level
Ji = index of branching facility chosen at level L
IJL = set of indices of unallocated facilities at level L
AEDil) = vector of average expected distances from region / to all other regions.
AX (I) = vector of average distance of region / to the new facilities that have been
currently located.
318
A. S. MARUCHECK AND A. A ALY
The parameters for creating and fathoming new nodes are:
KL = number of new nodes to be created at current level
NODE = counter for nodes created
ND = node number of the last node created at previous level
IP(L) = the new facility the branching facility at level L was allocated to according to the
node that was partitioned at level L
NL = number of new facilities at previous level
XX (J) = current location of new facility J
XLB(I) = lower bound at node /
Q(l) = the /th smallest value of mim>(/). w(k)}E[\R(j) - R(k)\] for all j < k
R(I) = the /th smallest value of 2Sw{j)[x2{j) - x\(j) + y2(J) ->T (./)].
STEP 0. Initialize the input parameters. (Compute upper and lower bounds on optimum.)
STEP 1. Let FX = oo.
I - 1
STEP 2. Arbitrarily allocate region / to new facility / - N
N
STEP 3. Solve the single facility location problem for all new facility XX (J), j = \ n
among the regions allocated to new facility j.
STEP 4. Evaluate F, the objective function value of the L — A problem, for the results of Step
3.
STEP 5. If FX — F > e, then replace FA'with F. Otherwise, go to 7.
STEP 6. For / = 1 A/, compute min{w(/) ■ E[\XX(j) - R (/){]}; let k be that facility
with the minimum expected value. Reallocate region /to new facility k. Go to 3.
STEP 7. Let z = FX.
v
STEP 8. Computer = .25 £ iv(/) • [x2(/) - .vl(/) + v2(/) - y\U)].
/=i
STEP 9. If z = z, stop. Go to 31.
(Initialize for level 1)
STEP 10. L = 1.
m
STEP 11. For / = 1 A/, compute AED(I) = £ E[\RU) ~ R(k)\}.
STEP 12. Let j\ = max AED(I). U] = {1. 2, . . . M\ - ./,.
STEP 13. Let NODE = 1. Assign region /] to new facility 1. Solve the location problem for
XX(\). Let NL = 1. Let IP(\) = 1.
(Advance to next level)
LOCATION- ALLOC ATION PROBLEM 3 1 9
STEP 14. LetL = L + 1.
(Compute Branching Facility)
i NL
STEP 15. Compute AXU) = -r*=- £ E[\R (/) - XXUI)\] for UIJL_X.
NL //=,
STEP 16. Let .4 = max AXU) and let IJL = A/L_, - ^.
(Create New Nodes)
STEP 17. Let AX = min(I, AO. Let ND = M?D£.
STEP 18. Create KL new nodes A7/) +1, . . . , ND + KL by allocation region jL to new facil-
ity 1 KL, respectively. Let NODE = ND + KL.
(Compute Lower Bounds on Nodes)
STEP 19. For node / = ND + 1 ND + KL, solve the location problem for the partial
allocation scheme: region ji allocated to new facility / — ND; jk allocated to IPik),
k = L — 1, L — 2, .. . , 1. Denote the objective function value LBU).
STEP 20. Compute the vectors QU) and R (/) using regions J, JeIJL.
M-L-N
STEP 21. If M- L - N > 5, let Qx = 1/2 £ QU). If M - L - N ^ 5, compute the
lower bound for the value M — L — N as given in Table 1 . Denote this value ()x
STEP 22. Let LBU) = LBU) + Qx, I = ND + 1 ND + k. If LBU) ^ z, fathom node
/.
STEP 23. Among the unfathomed nodes in 22, choose /* as the value of / such that
LBU*) = min LBU). If all nodes are fathomed, go to 27.
/
STEP 24. Let IP(L) = I* - ND.
STEP 25. If L < M, set NL and XX (j) j = 1, . . . , NL equal to the values found for /* in
Step 19 and go to 14.
STEP 26. If L = M, compare LBU*) to z. If LBU*) < z then z = LBU*). Fathom the
newly created nodes at level M.
(Backtracking Procedure)
STEP 27. Let L = L - 1. If L = 1, stop. Go to 31.
STEP 28. Consider all nodes / at level L that are unfathomed and have not been partitioned
such that their allocation scheme includes jL~\ allocated to IP(L - 1). If there is a
node /such that LBU) < z, go to 29. Otherwise, go to 27.
STEP 29. Choose /* such that LBU*) = min LBU) where /are the active nodes identified in
/
27. Let LL denote the new facility jL was allocated to at /'*. IP(L) = LL. Let NL
and ATO') become the appropriate values found in Step 19 for /*.
320
STEP 30. Go to Step 14.
A. S. MARUCHECK AND A. A. ALY
STEP 31. The optimal allocation scheme is the one associated with z, the optimal objective
function value.
6. VERIFICATION OF THE ALGORITHM
LABB, was coded in Fortran IV. The code was verified using an example problem
presented in Figure 1 where the w,-'s are 2,1,2,2,1, respectively, for the five regions.
10
— o
x,
2 A 6 8 10
FIGURE 1. A graph of an example problem
Both manual computation and the code produced the optimal allocation scheme to be: X\
serves regions 1 and 4 and X2 serves regions 2, 3 and 5. The two new facilities Xx and X2 were
located at (2.5,9) and (9.5,1.5), respectively. The optimal objective function value was 18.5.
The same problem with a centroid approximation produced a different allocation scheme:
A' | serves regions 3 and 5 and X2 serves regions 1, 2 and 4. The new facilities were located at
the points (4,8.5) and (9.5,1.5), the centroids of regions 4 and 3, respectively. These locations
used in the objective function involving the rectangular regions produced a value of 37, a 100
percent increase over the value for the optimal locations.
The impact of the sensitivity of the rectilinear distance metric to the centroid approach on
the location-allocation problem is serious; it has produced a nonoptimal allocation scheme and
inferior locations for the new facilities. As in the multifacility location problem, the centroid
approach does not even offer a good approximation to the solution of the location-allocation
model.
LOCATION-ALLOCATION PROBLEM
7. COMPUTATIONAL RESULTS
321
The computational results given in this section represent experience with the branch and
bound algorithm (LABB) for rectangular regions using a rectilinear distance metric. The prob-
lems were randomly generated from uniform distributions. All w(I)'s were generated from a
uniform [0,10] distribution. The xl(/)'s, x2(/)'s, ylUVs and ylUVs were each generated
from a uniform [0,100] distribution. All problems were run on an IBM 370/158J computer.
The results are summarized in Table 2.
TABLE 2 — Computational Results for Location-Allocation
Problems where n = 2, 3, and 4
m
No. of
Problems
Average
CPU Time
(seconds)
Average
No. of
Nodes
Created
Average
Maximum No. of
Active
Nodes
Average
Optimal
Node
n = 2
5
2
.585
11
1
10
6
3
1.01
27.3
2.7
19.7
7
4
1.00
25.75
3.5
20
9
3
1.99
94.33
4.67
79
11
3
5.37
240.3
8
119
15
2
8.55
330
11
219
20
2
11.33
332
18
39
25
2
19.08
349
23
69
30
1
33.02
517
28
59
35
1
51.15
541
33
69
n = 3
6
3
1.2
50
3.3
32
7
3
1.54
86
5.33
68
9
3
3.76
232
10.33
59
11
2
4.51
272
14
211.5
15
2
7.28
412.5
23
188
20
3
11.2
411
35.7
62.3
25
2
15.44
—
45
72
30
2
26.37
564
55
87
35
1
37.23
543
65
351
n = 4
7
3
2.73
129.33
7.3
87.33
9
3
3.01
223
11.3
118
11
2
4.8
316
23
38
15
2
6.73
416
36
54
20
4
11.77
586
48
74
25
1
13.26
458
66
94
Not surprisingly, the required computational time reflects the average number of nodes
created which, in turn, is a function of the size of the problem and the number of active nodes.
For the problems worked, no computational time was over one minute.
322 A. S. MARUCHECK AND A. A. ALY
In each of the cases of m for n = 2, the optimal allocation was examined to determine
what percentage of optimum was achieved by the lower bound at each level. In cases where m
was large, the general lower bound was used at the first m — 6 levels. The lower bound
improved rapidly from level to level; a typical improvement was ten percent of optimum. Usu-
ally, at the m — 6th level, the lower bound was within 85-90 percent of optimum. Thus, the
switch to the combinatorial lower bounds for the last five levels represented less improvement
from level to level, but convergence occurred rapidly.
The computational results in Table 2 indicate that the LABB algorithm obtains an optimal
solution for the L—A problem with rectangular regions in very reasonable time.
8. SUMMARY
In this paper the location-allocation problem for existing facilities uniformly distributed
over rectangular regions was considered. Previous works dealing with L — A systems were dis-
cussed, and the properties of the problem were developed. These properties indicated that
developing the optimal allocation scheme was the most important step in optimally solving the
L — A problem.
Computational results indicated that the exact algorithm (LABB) could obtain the optimal
solution for large problems in a reasonable time.
The branch and bound method (LABB) may be applied to location-allocation problems
with probability distributions on existing facilities other than uniform. Since the branch and
bound methods generate optimal allocation schemes no matter what type of objective function
is used, the only difference would be the way the location problems are solved at each node.
Solution techniques using other probability distributions are developed by Aly [1], Katz and
Cooper [8] and Wesolowsky [17]. It would be expected that the computational times to solve
these related problems would be similar to the times for the uniform distribution with adjust-
ments made on the basis of the speed of the individual solution technique.
A L — A problem may have constraints on the allocation scheme, on the locations of the
new facilities, or on both. In these cases, the constraints can be used as an additional test at
each node as a basis for fathoming the node.
REFERENCES
[1] Aly, A. A., "Probabilistic Formulation of Some Facility Location Problems," unpublished
Ph.D. dissertation, Virginia Polytechnic Institute, Blacksburg, Va. (1975).
[2] Aly, A. A. and White, J. A., "Probabilistic Formulation of the Multifacility Weber Problem,"
Naval Research Logistics Quarterly, 25, 531-547 (1978).
[3] Aly, A. A. and Steffen, A.E., "Multifacility Location Problem Among Rectangular
Regions," Working paper. School of Industrial Engineering, University of Oklahoma,
Norman, Okla. (1978).
[4] Bellman, R.E., "An Application of Dynamic Programming to Location-Allocation Prob-
lems," SI AM Review, 7, 126-128 (1965).
[5] Cooper, L., "Location-Allocation Problems," Operations Research, //, 331-334 (1963).
[6] Cooper, L., "Heuristic Methods for Location-Allocation Problems," SIAM Review, 6, 37-
53 (1964).
[7] Cooper, L., "Generalized Locational Equilibrium Models," Journal of Regional Science, 7,
1-17 (1967).
LOCATION-ALLOCATION PROBLEM 323
[8] Katz, I.N. and Cooper, L., "An Always Convergent Numerical Scheme for a Random
Locational Equilibrium Problem," SI AM Journal of Numerical Analysis, 17, 683-692
(1974).
[9] Kuenne, R.E. and Soland, R.M., "Exact and Approximate Solutions to the Multisource
Weber Problem," Mathematical Programming, 3, 193-209 (1972).
[10] Learner, E.E., "Locational Equilibrium," Journal of Regional Science, 8, 229-242 (1968).
[11] Love, R.F. and Morris, J.G., "A Computation Procedure for the Exact Solution of
Location-Allocation Problems with Rectangular Distances," Naval Research Logistics
Quarterly, 22, 441-453 (1975).
[12] Ostresh, L.M., "An Efficient Algorithm for Solving the Two Center Location-Allocation
Problem," Journal of Regional Science 15, 209-216 (1975).
[13] Rushton, G., Goodchild, M.F. and Ostresh, L.M., eds., Computer Programs for Location-
Allocation Problems, Monograph No. 6, Department of Geography, University of Iowa,
Iowa City, la. (1973).
[14] Sherali, A.D. and Shetty, CM., "The Rectilinear Distance Location-Allocation Problem,"
AIIE Transactions, 9, 136-143 (1977).
[15] Wendell, R.E. and Hurter, A. P., "Location Theory, Dominance and Convexity," Opera-
tions Research, 21, 314-320 (1973).
[16] Wesolowsky, G.O. and Love, R.F., "Location of Facilities with Rectangular Distances
Among Point and Area Destination," Naval Research Logistics Quarterly, 18, 83-90
(1971).
[17] Wesolowsky, G.O., "The Weber Problem with Rectangular Distances and Randomly Distri-
buted Destinations," Journal of Regional Science, 17, 53-59 (1977).
AN ITERATIVE ALGORITHM FOR THE
MULTIFACILITY MINIMAX LOCATION PROBLEM
WITH EUCLIDEAN DISTANCES
Christakis Charalambous
Department of Electrical Engineering
Concordia University
Montreal, Quebec, Canada
ABSTRACT
An iterative solution method is presented for solving the multifacility loca-
tion problem with Euclidean distances under the minimax criterion. The itera-
tive procedure is based on the transformation of the multifacility minimax
problem into a sequence of squared Euclidean minisum problems which have
analytical solutions. Computational experience with the new method is also
presented.
1. PROBLEM FORMULATION
To formulate the problem, let us suppose that m existing facilities are located at known
points (a\, b\), ia2, b2), ... , iam, bm) and n new facilities are to be located at points
ixh y{), ix2, yih ••• - (x„, y„). The cost
• -I ^
(la) /„(*, v,) = wu[(Xi - aj)2 + iy, - bj)2]V2, . ~ / 2' " ' \ "m
is incurred due to travel between new facility /' and existing facility j for all / and j(w,j is a non-
negative weight) and the cost
/= 1, 2 n-\
(lb) g,kixh y,, xk,yk) = v/A[U/ - xk)2 + (y, - yk)2Vn, k = { + ^ _ n
is incurred due to travel between new facilities / and k for all / < k (\tk is a nonnegative
weight).
From (la) and (lb) we can see that the maximum cost incurred due to movement
between facilities is:
(2) Fix, y) = max Uy(x,, y,), glk(xh yh xk, yk))
l<;<ffl
where
x = [xh x2, ... , xnV, y = \y\, y2, ■■■ > vj7".
The multifacility Euclidean minimax facility location problem is to find (x, y) which
minimizes Fix, y). The new facilities might be helicopter bases, transmitting stations where it
is desired to minimize the necessary signal, detection stations or civil defense sirens. An
interesting book in this area is that given in reference [8].
325
326
C. CHARALAMBOUS
One main characteristic of the objective function Fix, y) is that it has discontinuous par-
tial derivatives at points where two or more of the functions /y(x,-, ys), gikixh y/, xk, yk) are
equal to Fix, y). Various algorithms have been proposed for solving the general minimax
problem, some of the most relevant of which are due to Charalambous and Conn [2],
Charalambous [1], Dem'yanov and Malozemov [4], Madsen [11], and Dutta and Vidyasagar
[6]. More specialized algorithms for the minimax location problems were published by
Chatelon, Hearn and Lowe [3], Drezner and Wesolowsky [5], Elzinga, Hearn and Randolph
[7], and Love et al. [10].
In this paper we present a simple algorithm to minimize Fix, y). The original problem is
transformed into a sequence of unconstrained squared Euclidean minisum problems which have
analytical solutions. The resulting method is efficient and easy to implement on a computer.
Numerical results are presented which illustrate the usefulness of the new method to the mul-
tifacility location problem.
2. THEORETICAL RESULTS
LEMMA 1: The functions fjjixj.y,) and g^ix/, yh xk,yk) as defined in (la) and (lb)
respectively, are convex functions.
PROOF: See [9].
LEMMA 2: The function Fix, y) is continuous and convex.
PROOF: This follows from the fact that each f,jix,, y,) and gikixh y,, xk, yk) are con-
tinuous and convex functions (see for example [4]).
Let Pjjixj, y,) and q^ix,, y,, xk, yk) be the following 2«-dimensional column vectors:
in) )
(3a)
PijiXi, y.) =
/
in)
0
y, - ty
0
- (/)
(3b)
Qik(xh y,, xk, yk) =
— in + /)
(x, - xk)
ixk - X,)
0
iji ~ >'a)
(yk - yi)
- (/)
- ik)
in + I)
in + k)
MULTIFACILITY MINIMAX LOCATION 327
All the elements of ptJ are equal to zero except the rth and the in + /)th, and all the elements
of q,k are equal to zero except the /th, Ath, in + /)th and the in + k)th. Also note the
PijiXj, y,) and qlkixt, yt, xk, yk) are the gradient vectors of the following functions
\ [(xi-ajV+iyi-bj)2]
and
| [ix, - xk)2 + iy, - yk)2}
with respect to (x, y) respectively.
THEOREM 1 (Necessary and sufficient conditions for optimality): The necessary and
sufficient conditions for the point (x* y*) to be a minimum point for the function Fix, y) are
that there exist nonnegative multipliers X*(l= 1, 2 //, j= 1, 2, .... m),
fifk(l= 1, 2 n - 1, k = I + 1, ... , n) such that
(=1 y=l /t/V-*/» JV
»— 1 2 V/l
(4a) + £ Imi g , « '* y. vM tour. >•,*. ***, >■?) = o
/=1 A = /+l Slk^Xi, V/, XA, V^;
(4b) ZI^ + I I m,*a = 1
;=1 j=\ /=! A = /+l
(4c) a jCFOc* v*) - yjyxf, yf)) = 0,
(4d) /*f*(F(x* y*) - g,*(xf, J>f, xA*. y*k)) = 0
PROOF: The proof follows directly from the Kuhn-Tucker conditions for optimality for
this problem or from the Corollary of Theorem 3.2 of Dem'yanov and Malozemov [4]. Note
that \fj = /j.fk • = 0 for the functions fij(xif y,) and gik(xh y,, xk, yk) which are less than
Fix*, y*) at ix*, y*), i.e., for those functions which are not active at the solution (x* y*).
The A.,* and /xfk are called minimax multipliers. Also not that since f,,ix*, yf) = gik(xf, yf, xk,
yk) = Fix*, y*) from (4c) and (4d)) for corresponding kfj ^ 0 and fifk ^ 0, the denomina-
tors for all terms in the summations in (4a) can be replaced by 1 .
The possibility that some fyixf, yf) or gjkix*, yf, xk, y*) = 0 can occur. In this case
replacing the offending term by 6 > 0 will not change the optimality conditions since the asso-
ciated Lagrange multiplier will be zero for nontrivial problems.
Consider now the following problem (Euclidean-distance minisum location problem).
For given nonnegative values of A.,, = k,, and fxtk = JLlk
i = 1, 2 n
j = 1, 2 m
/= 1, 2, ...
, n - \
. . , n
minimize <$>ix, y, k, u.)
(x.y)
where
_ 1 n m _
(5) 4>(x, y k, JL) = | £ £ XtfwJICx, - a,)2 + (y, - bj)2}
+ T I I /SftVilCx) - x,)2 + (y, - yk)2}
1 l=\ k = l+\
k,j and/x/A are going to be called estimates for the minimax multipliers.
328 C. CHARALAMBOUS
Let
/= 1, 2 n
(6a) Wij = \jjW,jte 0) j= \t 2 m
/= 1, 2 n - 1
(6b) v/A =nikvik(> 0) fc- /+ 1, .... „.
THEOREM 2: For given nonnegative values of \y = Xy and ixlk = JLfk the function
4>(x, >>, X, JL) is convex.
PROOF: See [12].
THEOREM 3: If X„ = X* (/ = 1, 2 n, j = 1, 2 m), /ttft -/i£ (/ = I, 2,
..,« — 1, /c=/ + l «), the minimax multipliers corresponding to a minimum point
(x* >'*), then (x* >*) is a global minimum point of <I>(x, >\ A.*, /i*).
PROOF: The gradient vector of 4>(x, y, X *, /a*) at the point (x* >•*) is:
n— 1 n
i=l /=! /=1 fc-1+1
from Theorem 1 .
Since <Mx, .y, A. *, /x *) is a convex function the results follows.
Therefore, if we knew X* and /j.*, we could obtain (x*, y*) in one step by minimizing
<t>(x, >\ X *, fi*). Since we do not know these optimum multipliers in advance we need to esti-
mate them. Let (x, y) be a minimum point of <Mx, y, X, JL) for given values of X and Jl.
Define
(7a)
/= 1, 2 n
Xl^Xyf.jix,, y,)ls j=x 2 m
/= 1. 2 n - \
(7b) fifk = HikSikixi. y\. xk, yk)/s
k = I + 1 n
where
n-\ n
(8) s = II ^ijfiM, y,) + £ £ Hik8ik(xh yh xk, yk).
i=\ 7=1 /=1 *-/+]
Note that
(9a) \fj ^ 0, JZfk > 0
and
m _ n— 1
(9b) I I *,*,+ ! I i*|- 1.
/=! 7=1 /=1 k-l+l
Also at the point (x, y) the gradient vector of <Mx, y, X, Jl) must be zero. This gives us
n m _ yy.2
<*> ££xg- f 'J .Ptjix,, y,)
i=\ 7=1 /,;W/. #)
n— 1 n V/|.
+ L L M/a — ,_ _ _ — zt- 0/*(*/. J'/. xk, yk) = 0
/=i fc-/+i £/*(*/. >>/. **, >V
MULTIFACILITY MINIMAX LOCATION
329
which when compared with (4a, 4b) of Theorem 1 suggests XS, nfk are approximations to X*
/*/*■
THEOREM 4: At a minimum point ix, y) of <£>(x, y, X, fj.) the following inequality
holds:
where
F,ix, y) < Fix*, y*) < Fix, y)
n—l n
f/U>') = II *!//(/ (*/, ^/^ + Z Z /** &*(*/. ^ ■**• ^)
;=1 7=1 /=1 *=/+l
and X * and ZZ* are as they were defined in (7a) and (7b) respectively.
PROOF: The right hand side inequality is obvious. Also
Fix, y) ^ Fix*, y*) = min Fix, y)
(x,y)
= min
(x,y)
n m _ n—\n
ZZ^-^ + Z I m/UU y)
i=\ 7=1 /=1 fc=/+l
(since X * and /x * satisfy (9b))
^ min
Uv)
n m _ n—ltt
Z Z ><*jfiM> yJ + Z Z £?k8ik(xi, y,, xk, yk)
;=1 7=1 /=1 A=/+l
(from the definition of F(x, >>))
n-l
= 11 XJ//</U J') + Z Z H?k8ik(xi. yi> xk, yk), (from (9c))
;=1 7=1 /=1 fe=/+l
= F/Oc, JO.
3. THE ALGORITHM
The above theoretical results suggest the following algorithm:
STEP 1: Set r = 1
(r) _
1, /= 1, 2,
, n
j= 1, 2 m,
fiit'-l, 1=1,2 n-\ k=l + l,...,n.
STEP 2: Find the minimum point ix{r), y{r)) of <D(x, y, \{r), fM{r)). (See later for details).
STEP 3: At the point ix(r) , y(r)) calculate /y and glk and update X,7 and /u,/A. as follows: Set
^ = ii^^u(r,) + i z M^to^. ^ xp, j^>)
/=l 7=
X,
x (/■) /• (Y(r) (r)\
' « — - — r\ i = 1, 2, ... , n, j = 1, 2, . . . , m
Pik n /=1,2,..., (/7-1), A:=/+l n.
330
STEP 4: Calculate
C. CHARALAMBOUS
,= 1 y-i
n-1
+ 11 vrl}g*(xM.yr.xkM.yi'))-
l=\ k=l+\
STEP 5: Stopping criterion: If (F(x{r\ y(r)) - F,{x[r) , yir)))l F{xir) , y{r)) < e stop; Otherwise
set r *— r + 1 and go back to Step 2. (e is a prescribed tolerance).
3.1 Finding the Optimum Solution of the Quadratic Function
For given nonnegative values of A,, and Jx/k we want to find the minimizing point (3c, y)
for<J>0c, y, \, JL). Let
(10a,b)
(11)
a =
Z w\jaj
7-1
Z *!&
7=1
Z wnj<*j
L 7=1
b =
A =
0i
-Vl2
"Vl2
02
— V
I n
v2„
Z »u^
7=1
Z W'27 A
7=1
Z Wnjbj
L 7=1
■v2„
0„ J
(12)
^(=I^ + Ivy, /= 1, 2 R
7=1 7=1
Then the optimum solution can be obtained by solving the two systems of equations (see,
for example, [8]).
(13a,b) Ax = a and Ay = b.
Since for given nonnegative values of \y and /*ik the function <t>(x, y, \, /x) is convex, it fol-
lows that its Hessian matrix A is positive semi-definite. Also using the fact that A is symmetric
we can write
MULTIFACILITY MINIM AX LOCATION 331
(14) A = LLT
where L is an n x n lower triangular matrix.
This is called the Cholesky decomposition of A and requires about a;3/ 6 multiplications.
By using (14) for each right hand side of (13), solve the following system to obtain 3c and y.
Lp = a Lq = b
LTx = p LTy = q.
This requires about 2(n2 + n) multiplications.
Note that if /3, = 0, then the /th row of A, the /th column of A, a, and b, are all equal to
zero, and can be removed in solving for x and y. In this case we have infinite solutions for the
location 6c,, y,-) of the /th facility.
4. NUMERICAL EXAMPLES
We give a number of numerical examples to illustrate the usefulness of this approach to
solving multifacility location minimax problems. For all the examples considered e = 10~4.
Computations were carried out at Concordia University on CDC 64000 computer using single
precision arithmetic. A user-oriented computer program written in Fortran IV implementing
the above algorithm is available from the author.
EXAMPLE 1: Love, Wesolowsky and Kraemer [10], considered the problem where
n = 2, m = 5 and (ah 6,) = (39.12, 28.11), ia2, b2) = (39.50, 28.28), ia3, b3) = (37.88,
29.87), (fl4f b4) = (38.59, 27.03), (a5, b5) = (38.38, 30.28), v12 = 1, and
W = (wil) =
14 4 4 1
4 1114
The results obtained by using the present approach are summarized below:
Results for Example 1
Number of Iterations
1
2
6
10
30
40
Fix, y)
F,ix, y)
6.5237
4.3311
5.9768
5.0112
5.9138
5.5857
5.8738
5.7367
5.8554
5.8526
5.85496
5.85439
Values of (Xy), (/*/*), (/#), igik) and ix, y) after 40 iterations:
0. 0.00047 0.49982 0.49965 0.
0.00003 0. 0. 0. 0.0003
A = (A„) =
AM 2
= 0.
W* 30)-
xi = 38.2356, x2
0.9476 5.1031 5.85466 5.85496 1.8355
4.58541 1.1831 1.1011 2.1709 4.58541
gn = 0.9052
yx = 28.4502, y2= 29.1950.
It can be seen that only functions /13 and f]4 define the minimax function at the solution
point and A,, — - 0, for all (/, /) except \13 and A.]4. Also /x\2 = 0. In other words the X(/ and
332
C. CHARALAMBOUS
fXjk corresponding to functions fl} and glk that are not active at the minimax solution tend to
zero, as it should be expected. Let
h = (U J)\fu(x y) < 0-99 F,ix, y), \tJ < 10-2}
h = Ui k)\glk(x, y) < 0.99 F,ix, y), /x/A < 10"2}
where (x, y) is the minimum point obtained at the end of the 40th iteration. If
(5c, y) = (x*, y*), then the elements of IK and /M will correspond to functions which are not
active at (x* y*) and A* ■ = 0, (/', j) € IK /x *A = 0, (/, k) € /M. Also, if (x, y) is in the neigh-
borhood of (x*, y*), then most likely the elements of IK and /M will correspond to functions
which are not active at the solution.
By excluding from the problem the functions corresponding to the elements belonging in
IK and /M, using the values of A.,, and ixlk obtained at the end of the 40th iteration for the
remaining functions, the present algorithm reached the exact solution to the problem in two
additional iterations. The final results obtained are summarized below. The method required
0.68 sec CPU time to reach the final results shown. From now on this additional part of the
algorithm will be called Phase 2, and the original part of the algorithm where all functions are
considered (algorithm in Section 3) will be called Phase /:
Final Results for Example 1: Fix*, y*) = F,ix* y*) = 5.85481, A * = 0. except Af3 =
Af4= 0.5,/tf2= 0.
(f.jix*. ym))
0.9481 5.1055 5.85481 5.85481 1.8357
4.58541 1.1831 1.1011 2.1709 4.58541
gu(x*. y*) = 0.9057
xf= 38.235 y*= 28.45
x*2= 38.75 v2*= 29.195
Exact solution.
Since /n and f\4 are the only functions defining the minimax solution and both of them
depend only on (x|, j>i) (i.e., they are independent of the value of (x2, ^2))- The value of
(x*> J 2) ^ not unique, but the value of (x*( vf) is unique. In fact, (xf, y*) is any point in the
set
5,2) = n sj
7-1
(2)
where
5-(2)={(x y)\w2J[(x - aj)2 + (y - ^)2]1/2 < Fix*, y*)}, j = 1, 2 5
S£2) = {(x, v)|v12[(x - xf)2 +(y- vf)2]1/2 < Fix*, y*)}.
The boundary of the set 5/2) is a circle with center {aJt bj) and radius Fix*, y*)/w2n for
j = 1, 2 5 and center (x*, >•*) and radius Fix*, y*)hn for j = 6. For this particular
example the solution set St2) for (x2> y*) is given by the intersection of the sets S[2) and Ss2).
This is illustrated in Figure 1. Since the value of (x2> y*) is not unique and our interest is on
the minimax facility location problem it would be appropriate to choose the position of the
second new facility such that the function
^2(^2. ^2) = max {f2j(x2, v2), £i2(x2> y2, xf, y*)}
is minimized in the set S(2). The optimum solution to this problem occurs at the point C2, and
coincides with the minimum point obtained by using the present algorithm. In this case f2\ and
/25 define the minimax solution.
MULTIFACILITY MINIMAX LOCATION
333
•CD " Location of the ith
existing facility
x\ij - Optimum location of
new facility i.
Figure 1. Illustration of solution set for example 1.
The Revised Algorithm
In summary the revised algorithm operates in two phases:
(i) Use Phase 1 (algorithm in Section 3) with e = 10 4 to get to the neighborhood of
the solution and to identify the functions that are inactive at the solution.
(ii) Continue the algorithm by using Phase 2 where the inactive functions are
excluded from any further consideration.
EXAMPLE 2: In this case n = 3, m = 5.
334
C. CHARALAMBOUS
i
1
2
3
4
5
a,
0
2
5
7
8
b,
0
8
4
6
2
H/ =
6 12 0 0
0 0 13 4
0 5 2 0 2
'12
= 0,
'13
= 2, V23-1.
The results obtained by using Phase 1 of the algorithm are summarized below. It can be
seen that only functions fa and f^ define the minimax function at the solution point, and
\jj — * 0, for all (/', j) except A32 and \35. Also
IjLik — 0, L < / < k < 3.
Results for Example 2 using Phase 1 of the Algorithm
Number of Iterations
1
10
20
30
40
59
Fix, y)
F,ix, y)
13.0004
09.0075
12.1901
11.7768
12.1319
12.0475
12.1242
12.0996
12.1225
12.1143
12.1219
12.1206
Values of U„), (/x/aK C/y)» Of/*) and ix, y) after 59 iterations
0.0002 0. 0. 0. 0.
0. 0. 0. 0. 0.
0. 0.2855 0. 0. 0.7136
A
M =
Hn = Ml3 = M23 = 0.
(fu(x,y)) =
10.9488 6.5178 9.4821 0. 0.
0. 0. 2.5873 7.0682 07.0682
0. 12.1216 5.2442 0. 12.1219
x, = 0.92850, x2 = 7.57143, x3 = 3.71311
y] = 1.57092, ?2 = 3.71429, y3 = 6.28460.
gi2=0, gu= 10.9495
g23= 4.6361
Starting f:om the results obtained at the end of the 59 iterations of Phase 1 and using
Phase 2, the algorithm reached the exact solution to the problem in two additional iterations.
The final results obtained are summarized below. The method required 0.85 sec CPU time to
reach the final results shown.
Final Results for Example 2: Fix*, y*) = F{(x*, y*) = 12.1218305, \* = 0., except
\*2= 0.28571, A 3*5= 0.71429, /u £ = 0., V(/, k).
WiAx*. y*)) =
10.949 6.5177 9.4821 0. 0.
0. 0. 2.5873 7.06818 07.06818
0. 12.1218 5.2450 0. 12.1218
gnix*, y*) = 0, gliix*. y*) = 10.953, ^23(x*. y*) = 4.6357
MULTIFACILITY MINIMAX LOCATION
335
xf = 0.92850 >>f== 1.57091
x* = 7.57143 v2*= 3.71429
x3*= 3.71429 j>3*= 6.28571
Exact solution.
Since f^ and f^ are the only functions defining the minimax .solution and both of them
depend only on (x3, y3) the values of (xf, y*) and ix*, y*) are not unique, but the value of
(x*, y*) is unique. In fact (xf, y*) in any point in the set
:d) ^
n s
ID
j=i
and ix*, .yf) is any point in the set
5(2) = p 5.(2)
7=1
where
(15)
5,(/) = (U y)\wu(x - aj)2 + iy - bj)2]l/2 < Fix*, y*)},
;= 1, 2, 7= 1, 2 5
56(,) = {(x, y)\vu(x - x*)2 + (y - y*)2Vn ^ Fix*, y*)}, i = 1, 2.
The solution sets are illustrated in Figure 2.
*Ci) ~ L°cation of the ith
existing facility
xQJ - Optimum location of
ith new facility
Figure 2. Illustration of solution sets for example 2.
336 C. CHARALAMBOUS
As in Example 1, it would be appropriate to choose the position of the first and the
second new facilities such that the function
Fli2Ui, yx, x2, y2) = max {f.jix,, y,), g,3(xh y,, xf, j|), gn(xu yh x2, y2))
K / < 2
l</<2
is minimized, subject to the conditions (x1( yx) € S<n and (x2, y2) € S{2). Since v12 = 0 func-
tion £12 can be excluded in defining function Fx 2. The optimum solution to this problem is
such that (xf, y*) is unique and is that obtained by using the present algorithm (Point C\ in
Figure 2), and (x*, y2) in any point in the set 5<2).
Again it would be appropriate to choose the position of the second new facility such that
the function
F2(x2, y2) = max {f2j(x2, y2), g2x{x2, y2, xf, >f). g2i(x2, ^2- *3- >'*)}
is minimized. The optimum solution to this problem occurs at point C2 in Figure 2 and is that
obtained by the present algorithm.
5. CONCLUSIONS
An algorithm for the minimax facility location problem using Euclidean distances was pro-
posed. Although no proof of convergence of the algorithm is available, for all examples con-
sidered, the algorithm converged to a minimax solution. Since there is no line search in the
algorithm it follows that one iteration is the same as one function evaluation.
6. ACKNOWLEDGMENT
The author wishes to thank P. Lafoyiannis for programming the present algorithm, and
Paul Calamai for the useful criticisms. Furthermore, the author wishes to thank the referee for
his valuable comments. This work was supported by the Natural Sciences and Engineering
Research Council of Canada.
REFERENCES
[1] Charalambous, C, "Acceleration of the Least /rth Algorithm for Minimax Optimization
with Engineering Applications," Mathematical Programming, /7, 270-297 (1979).
[2] Charalambous, C. and A.R. Conn, "An Efficient Method to Solve the Minimax Problem
Directly," SIAM Journal of Numerical Analysis, 15, 162-187 (1978).
[3] Chatelon, J. A., D.W. Hearn and T.J. Lowe, "A Subgradient Algorithm for Certain
Minimax and Minisum Problems," Mathematical Programming, 15, 130-145 (1978).
[4] DenVyanov V.F. and V.N. Malozemov, "Introduction to Minimax," (John Wiley & Sons,
New York, N.Y. 1974).
[5] Drezner Z. and G.O. Wesolowsky, "A New Method for the Multifacility Minimax Location
Problem," The Journal of the Operational Research Society, 29, 1095-1101 (1978).
[6] Dutta S.R.K. and M. Vidyasagar, "New Algorithms for Constrained Minimax Optimiza-
tion," Mathematical Programming, /.?, 140-155 (1977).
[7] Elzinga, J., D.W. Hearn and W.D. Randolph, "Minimax Multifacility Location with
Euclidean Distances," Transportation Science, 10, 321-336 (1976).
[8] Francis, R.L. and J. A. White, "Facility Layout and Location: An Analytic Approach,"
(Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974).
MULTIFACILITY MINIMAX LOCATION 337
[9] Love, R.F., "Locating Facilities in Three-Dimensional Space by Convex Programming,"
Naval Research Logistics Quarterly, /6, 503-516 (1969).
[10] Love, R.F., G.O. Wesolowsky and S.A. Kraemer, "A Multifacility Minimax Location
Method for Euclidean Distances," International Journal of Production Research, //,
37-45 (1973).
[11] Madsen, K., "An Algorithm for Minimax Solution of Overdetermined Systems of Non-
Linear Equations," Journal of the Institute of Mathematics and its Applications, /6,
321-328 (1975).
[12] White, J. A., "A Quadratic Facility Location Problem," AIIE Transactions, J, 156-157
(1971).
COUNTEREXAMPLES TO OPTIMAL PERMUTATION
SCHEDULES FOR CERTAIN FLOW SHOP PROBLEMS
S. S. Panwalkar, M. L. Smith
Department of Industrial Engineering
Texas Tech University
Lubbock, Texas
C. R. Woollam
Department of Management
The University of Tennessee
Knoxville, Tennessee
ABSTRACT
It is well known that a minimal makespan permutation sequence exists for
the n x 3 flow shop problem and for the n x m flow shop problem with no in-
process waiting when processing times for both types of problems are positive.
It is shown in this paper that when the assumption of positive processing times
is relaxed to include nonnegative processing times, optimality of permutation
schedules cannot be guaranteed.
1. INTRODUCTION
Consider the n job-m machine flow shop sequencing problem in which processing times
are nonnegative. In the following we will show that a permutation schedule may not be optimal
for the classical flow shop problem involving three machines and for the n x m flow shop prob-
lem with the no in-process waiting constraint. We will use the 4x3 problem data shown in
Table 1 and the nonpermutation schedule P defined in Table 2. Note that job 2 does not
require processing on machine B.
TABLE 1 — Processing
Time Matrix
Job
Machine
1
2
3
4
A
B
C
1
2
4
2
6
0
1
3
3
4
3
1
TABLE 2 — Nonpermutation
Schedule P
Machine
Job Order
A
B
C
1,2,3,4
1,3,4
2,1,3,4
339
340 S. S. PANWALKER, M. L. SMITH AND C. R. WOOLLAM
2. THREE MACHINE FLOW SHOP PROBLEM
In [8], Johnson proved the optimality of a permutation schedule for the /; x 2 problem
under the assumption of positive processing times. He then extended the results to the «x3
problem and proved that an optimal permutation schedule exists. A number of researchers [3,
9, l-p.9, 2-p.l36, 4-p.84, 5-p.343, 6-p.201] have since relaxed the assumption of positive pro-
cessing times to nonnegative ones. It is easy to verify that for the problem in Table 1, an
optimal permutation schedule has a makespan of 16 units while the nonpermutation schedule P
defined above has a makespan equal to 14 units.
3. FLOW SHOP PROBLEM WITH NO IN-PROCESS WAITING
We now consider the n x m flow shop sequencing problem with no in-process waiting
allowed [10, 11]. In [11], Wismer considers nonnegative processing times. However, he
allowed only permutation schedules. In [2], Baker recognized the fact that a nonpermutation
schedule may be optimal when processing times are nonnegative. Gupta [7], on the other
hand, has proved (Theorem 1) that even when the processing times are nonnegative only per-
mutation schedules are feasible. The example in Table 1 is a counterexample to Gupta's
theorem. For the no waiting problem, the best permutation sequence has a makespan of 17 as
opposed to sequence P which has a makespan of 15. It may be noted that in both cases above,
the minimum problem size needed to obtain a better nonpermutation schedule is 3 x 3.
REFERENCES
[1] Ashour, S., Sequencing Theory (Springer- Verlag, New York, N.Y., 1972).
[2] Baker, K.R., Introduction to Sequencing and Scheduling (John Wiley & Sons, Inc., New
York, N.Y., 1974).
[3] Burns, F. and J. Rooker, "A Special Case of the 3 x n Flow-Shop Problem," Naval
Research Logistics Quarterly, 22, 811-817 (1975).
[4] Conway, R.W., W.L. Maxwell and L.W. Miller, Theory of Scheduling (Addison-Wesley
Publishing Co., Reading, Mass. 1967).
[5] Eiselt, HA. and H. Von Frajer, Editors, Operations Research Handbook (Walter de Gruyter
and Co., Berlin, 1977).
[6] Fabricki, W.J., P.M. Ghare and P.E. Torgersen, Industrial Operations Research (Prentice-
Hall, Inc., Englewood Cliffs, N.J., 1972).
[7] Gupta, J.N.D., "Optimal Flowshop with No Intermediate Storage Space," Naval Research
Logistics Quarterly, 23, 235-243 (1976).
[8] Johnson, S.M., "Optimal Two- and Three-State Production Schedules with Setup Times
Included," Naval Research Logistics Quarterly, /, 61-68 (1954).
[9] Lomnicki, Z.A., "A Branch-and-Bound Algorithm for the Exact Solution of the Three-
Machine Scheduling Problem," Operational Research Quarterly, 76, 89-100 (1965).
[10] Reddi, S.S. and C.V. Ramamoorthy, "On the Flow Shop Sequencing Problem with No Wait
in Process," Operational Research Quarterly, 23, 323-331 (1972).
[11] Wismer, D.A., "Solution of the Flowshop Scheduling Problem with No Intermediate
Queues." Operations Research, 20, 689-697 (1972).
A NOTE ON
A MAXIMIN DISPOSAL POLICY UNDER NWUE PRICING*
Manish C. Bhattacharjee
Indian Institute of Management
Calcutta, India
ABSTRACT
For the classical disposal model for selling an asset with unknown price dis-
tribution which is NWUE (new worse than used in expectation) with a given
finite mean price, this note derives a policy which is maximin. The gain in us-
ing the maximin policy relative to the option of selling right away is convex de-
creasing in the continuation cost to mean price ratio. The relevant results of
Derman, Lieberman and Ross also follow as a consequence of our analysis.
Our theorem provides a practical justification of their main result on the cutoff
bid for the disposal model subject to NWUE pricing.
1. INTRODUCTION
Consider an indivisible asset for which offers come in sequentially, with a continuation
cost c > 0 for each day the bid is not accepted. When the successive offers are independent
identically distributed with a distribution F, this classic disposal model has been reconsidered by
Derman, Lieberman and Ross [3] in an adaptive setting and when F is NWUE (new worse than
used in expectation). While a complete solution is given in the adaptive case, their main result
in the other case provides a lower bound on the optimal cutoff bid which, except for implying a
corresponding lower bound on the optimal return (viz. Theorem 1 and Proposition 2 in [3]), is
of limited practical value if F is NWUE but unknown.
The purpose of this note is to show that when the pricing is NWUE with a given mean
price but is otherwise unknown, there is a maximin disposal policy determined by the lower
bound for the cutoff bid given in [3]. As a by-product of our analysis, the Derman-
Lieberman-Ross results on the cutoff bid also follow directly without invoking the ordering
relationship among distributions defined through integrals of increasing convex functions as
considered in [3].
2. MAXIMIN POLICY UNDER NWUE PRICING
Let F = 1 — F. For the classic disposal model [2], [3], with F known, recall there is an
optimal policy maximizing expected return— which accepts offer x if and only if x ^ Xf, and
has return (xF + c), where the optimal cutoff bid xF is given by
This research was supported by the Center for Management Development Studies at the Indian Institute of Manage-
ment, Calcutta, India under research project 441/CMDS-APRP-I.
341
342 M. C. BHATTACHARJEE
(2.1) xf=inf
z : z
> \f~y dF-c\/ F(z)
= inf {z : c ^ £f(AT- z)+) ,
Z ^ 0 being distributed as F. Let xexp denote the optimal cutoff bid for an exponential price
distribution with the same mean as that of F, this distribution being henceforth abbreviated as
'exp\ Then
(2.2) xexp = — m log (c/m), where m = J Fiy) dy > c.
Let
(2.3) LFix) = E max iX.x — c) — x.
Note Z./ (c -f x) = E(X — x)+ - c; thus (2.1), when Fis continuous, implies Lh (c + xF) = 0.
F(y) dy.
Let tt denote any policy (including randomized ones with past memory) and R(n,F) its
return. For any x, let w(x) be the (stationary nonrandomized) policy which sells as soon as a
bid of amount x or more is received. For any x such that F(x) < 1, the return R (x,F) of the
policy 77 (x) is:
oo
(2.4) R(x.F) = X {£<*!* > x)- (n - \)c) F" ' (x) F(x)
= £f(A-|A- ^ x)
1 - Fix)
= x + [EF(X - x)+ - cF(x)]/F(x)
= x + c + [LF(c + x)lF(x)].
Now suppose the pricing distribution F with mean m < oo has the NWUE property [1]
defined by
f F(y) dy ^ m Fix)
i.e., infjf^o EFiX — x\X ^ x) = EFX. Then we have the following:
THEOREM: Suppose the price distribution F is NWUE and the continuation cost
c < m = EFX < oo. If we only know the mean m (and not /•), then the policy which sells as
soon as the offered price is xcxp or more is maximin.
To prove the theorem, we will use the following generalization of a result (lemma 6.4, p.
112) in [1], a direct application of which yields Proposition 2 and Theorem 1 of [3].
LEMMA: If Fis NWUE with mean m < °° and <f> iy) is nondecreasing on (0,00), then
Jo 0 iy) Fiy) dy > JQ <f> (y) e~y/mdy.
If Fis NBUE (new better than used in expectation), the inequality is reversed.
NOTE ON MAXIMIN DISPOSAL POLICY 343
def C x —
PROOF: Let Ybe a random variable distributed as TF(x) = m~x F(y) dy and let Z
be exponential with mean m. Now F \s NWUE implies the inequality F(y) dy ^ me xlm
(viz., [1], p. 187), i.e., Z is stochastically smaller than Y. Hence,
I 0(v)F(v) dy= m I 0(y) TF(dy)
•* o ^ o
= m E<t>(Y) > mE<f>(Z)= I <(t(y) e~y/m dy.
•'0
The NBUE case (EF(X — x\X ^ x) < £,/-Ar) follows by reversing all inequalities.
PROOF of Theorem: For any x > 0, choose 0, in the lemma, as the indicator of [x,°°)
to conclude
(2.5) c + LF(c + x) = f™ F{y) dy > J"°° e~-v/m dy = c + Lexp(c + x),
when F is NWUE. Thus, LF(c + x) ^ Lexp(c + x). This with (2.1) implies that xF ^ xexp,
as in Derman, Lieberman and Ross [3]. Hence, when Fis NWUE, by (2.4) we have
(2.6) R(xexp,F) ^ c +xexp,
since LF(c + xexp) ^ Lexp(c + xexp) = 0-. where the inequality is due to the NWUE hypothesis
and the last equality holds by continuity of the exponential distribution. Also, for any F,
(2.7) supnR(ir,F) = c + xF = R(xF,F),
since the policy it (xf) has the maximal return for a given price distribution F. Hence, using
(2.5), (2.6) and (2.7), and infF denoting infimum over all NWUE distributions Fwith a given
mean m, we have
c + xexp ^ inf/r R (xexp,F) ^ supx inff R (x,F)
< sup„. inf/r R(n ,F)
< inf/r sup„. R (it ,F)
< supjr R (7r,exp)
= R (xexp> exp) = c + xexp.
Thus, R(xexp, exp) = supff inf/r R(tt ,F) and the policy it (xexp) is maximin, i.e., it maximizes
the reward from the worst possible NWUE law with given mean.
REMARKS:
1. Note, (2.5) together with (2.1) implies Proposition 2 of [3], by arguments paralleling
those leading to (2.6). Likewise, the main result (Theorem 1) of [3] for NWUE pricing is con-
tained in the proof of our Theorem.
2. The maximin policy behaves as if the price distribution, with known mean m, is
exponential. Its relative gain compared to selling right away is
m~x R (xexp, exp) - 1 = -(1 - a) - loge a > 0,
where a = c/m, the continuation cost to mean price ratio; 0 < a < 1. The relative gain
increases as a decreases.
3. Suppose the price distribution F is arbitrary but strictly increasing and let £ j be the
2
median price. Then we will show:
344 M. C. BHATTACHARJEE
(2.8) xF > (£x - 2c)+.
2
Take c < — £ 1 . If (2.8) does not hold, then xF < (£ j - 2c) and using (2.4) and (2.7), we
2 1 1
have
c + xf ^ R (2c + X/r.T7) > 2c + xF - c {File + xF)/Fi2c + xF)} > c + xF ,
a contradiction. When the price distribution is NWUE, a bound stronger than (2.8) actually
holds. To see this, note that if Fis NWUE with mean m, then using (2.4) we get
(2.9) Rix.F) = x + EF(X - x\X > x) - c {Fix)/ Fix)}
^ x + m - c {Fix)/ Fix)}
for all x such that Fix) < 1. Accordingly,
c + xF = R ixF,F) ^ R (£±I F) > £j_ + m - c,
'2 2
where the first inequality is due to (2.7) and the next one follows from (2.9). Hence,
Xf. ^ m + £ | — 2c > £ ] — 2c and (2.8) holds afortiori. Since xF is nonnegative and a > b
T 1
implies a+ ^ b+ ', the resulting inequality xf ^ (m + £ | — 2c)+ is a sharpening of (2.8).
I
ACKNOWLEDGMENT
Thanks are due to the referee for helpful comments.
Note added in proof. Bengt Klefsjo, in a private communication, has recently pointed out to the
author that our results (main theorem and remarks) remain valid for the broader class of
HNWUE (harmonic new worse than used in expectation) price distributions. The classes
HNWUE (HNBUE) which are less well known, were introduced by Rolski [5] and further stu-
died by Klefsjo [4], are strictly bigger than NWUE (NBUE). A life distribution Twith mean m
is said to be HNWUE (HNBUE) if
J.oo _
Fiy)dy > me-xlm.
The reason for the name HNWUE (HNBUE) derives from the fact that (2.10) is equivalent [4]
to
i-i
- C {EFiX- y\X > y))-xdy
' ** U
^ m= EFX.
It can be easily seen that the Lemma remains true under HNWUE (HNBUE) hypothesis and
hence our results carry over to HNWUE price distributions.
REFERENCES
[1] Barlow, R. and F. Proschan, Statistical Theory of Reliability and Life Testing Probability
Models (Holt, Rinehart and Winston, New York, N.Y., 1975).
[2] Chow, Y.S. and H. Robbins, "A Martingale Systems Theorem and Applications," in
Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, /,
93-104 (1961).
[3] Derman, C, G.J. Lieberman and S.M. Ross, "Adaptive Disposal Models," Naval Research
Logistics Quarterly, 26, 33-40 (1979).
NOTE ON MAXIMIN DISPOSAL POLICY 345
[4] Klefsjo, B., "Some Properties of the HNBUE and HNWUE classes of Life Distributions,"
Statistical Research Report #1980-8, University of Umea, Sweden (1980).
[5] Rolski, T., "Mean Residual Life," Proceedings of the 40th session, Bulletin of the Int'l Stat.
Inst., Voorburg, Netherlands, 4, 266-270 (1975).
INFORMATION FOR CONTRIBUTORS
The NAVAL RESEARCH LOGISTICS QUARTERLY is devoted to the dissemination of
scientific information in logistics and will publish research and expository papers, including those
in certain areas of mathematics, statistics, and economics, relevant to the over-all effort to improve
the efficiency and effectiveness of logistics operations.
Manuscripts and other items for publication should be sent to The Managing Editor, NAVAL
RESEARCH LOGISTICS QUARTERLY, Office of Naval Research, Arlington, Va. 22217.
Each manuscript which is considered to be suitable material tor the QUARTERLY is sent to one
or more referees.
Manuscripts submitted for publication should be typewritten, double-spaced, and the author
should retain a copy. Refereeing may be expedited if an extra copy of the manuscript is submitted
with the original.
A short abstract (not over 400 words) should accompany each manuscript. This will appear
at the head of the published paper in the QUARTERLY.
There is no authorization for compensation to authors for papers which have been accepted
for publication. Authors will receive 250 reprints of their published papers.
Readers are invited to submit to the Managing Editor items of general interest in the field
of logistics, for possible publication in the NEWS AND MEMORANDA or NOTES sections
of the QUARTERLY.
NAVAL RESEARCH
LOGISTICS
QUARTERLY
JUNE 1981
VOL. 28, NO. 2
NAVSO P-1278
CONTENTS
ARTICLES
Applications of Renewal Theory in Analysis
of the Free-Replacement Warranty
Comparing Alternating Renewal Processes
Shock Models with Phase Type
Survival and Shock Resistance
An Early-Accept Modification to the Test
Plans of Military Standard 781C
A Two-State System with Partial
Availability in the Failed State
An Analysis of Single Item Inventory
Systems with Returns
Analytic Approximations for (s,S) Inventory
Policy Operating Characteristics
Optimal Ordering Policies When Anticipating
Parameter Changes in EOQ Systems
Systems Defense Games:
Colonel Blotto, Command and Control
On Nonpreemptive Strategies in
Stochastic Scheduling
Postoptimality Analysis in Nonlinear Integer
Programming: the Right-Hand Side Case
An Efficient Algorithm for the Location-
Allocation Problem with Rectangular Regions
An Iterative Algorithm for the Multifacility
Minimax Location Problem
with Euclidean Distances
Counterexamples to Optimal Permutation
Schedules for Certain Flow Shop Problems
A Note on a Maximin Disposal Policy
Under NWUE Pricing
Page
W.R. BLISCHKE 193
E.M. SCHEUER
D.T. CHIANG 207
S.-C. NIU
M.F. NEUTS 213
M.C. BHATTACHARJEE
D.A. BUTLER 221
G.J.LIEBERMAN
L.A. BAXTER 231
J.A. MUCKSTADT 237
M.H. ISAAC
R. EHRHARDT 255
B. LEV 267
H.J. WEISS
A.L. SOYSTER
M. SHUBIK 281
R.J. WEBER
K.D. GLAZEBROOK 289
M.W. COOPER 301
A.S. MARUCHECK 309
A.A. ALY
C. CHARALAMBOUS 325
S.S. PANWALKAR 339
M.L. SMITH
C.R. WOOLLAM
M.C. BHATTACHARJEE 341
OFFICE OF NAVAL RESEARCH
Arlington, Va. 22217
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